Calc 3 Part 3 Review
What is a gradient field, how to find it, draw it, interpret it, and how it relates to level curves?
*Definition*: A gradient field is a vector field ∇f associated with a scalar function f(x,y,z). *Finding*: Compute the partial derivatives ∂f/∂x, ∂f/∂y, ∂f/∂z & form the vector ∇f = ⟨∂f/∂x, ∂f/∂y, ∂f/∂z⟩. *Drawing*: At each point, draw the vector ∇f pointing in the direction of the steepest increase of f. *Interpreting*: The magnitude of ∇f represents the rate of change of f at that point. *Relation to Level Curves*: ∇f is always orthogonal/perpendicular to the level curves of f.
How to evaluate triple integrals in cylindrical coordinates?
1.) Express the function f(x,y,z) in terms of r,θ,z. 2.) Identify the bounds for r, θ, and z. 3.) Replace x and y with rcos(θ) and rsin(θ), respectively. 4.) Replace dA with rdrdθ 5.) Replace dz with dz. 6.) Evaluate the resulting integral. ∭E f(x,y,z) dV=∭E f(rcos(θ),rsin(θ),z)⋅rdrdθdz
How to evaluate triple integrals in Spherical Coordinates?
1.) Express the function f(x,y,z) in terms of ρ,θ,ϕ. 2.) Identify the bounds for ρ, θ, and ϕ. 3.) Replace x, y, and z with the spherical coordinate equations. 4.) Replace dV with ρ2sin(ϕ)dρdθdϕ. 5.) Evaluate the resulting integral. ∭E f(x,y,z) dV=∭E f(ρsin(ϕ)cos(θ),ρsin(ϕ)sin(θ),ρcos(ϕ))⋅ρ2sin(ϕ)dρdθdϕ
How to use the change of variable to evaluate double and triple integrals?
1.) Given Integral: Start with an integral over a region D in the original coordinate system. 2.) Transformation T: Apply a change of variable using a transformation T to switch to a new coordinate system. 3.) Jacobian Determinant: Compute the Jacobian determinant ∣JT∣ for the transformation. 4.)New Integral: Rewrite the integral in terms of the new variables using the Jacobian determinant. ∬D f(x,y) dA =∬R f(u,v)⋅∣JT∣dudv
How to rewrite the region using the transformation?
1.) Given Region D: Express the region D in terms of the original variables. 2.)Transformation T: Apply the transformation T to map points from D to a new region R in another coordinate system. 3.)New Region R: Express the region R in terms of the new variables. (u,v)=T(x,y)
How to use transformations to convert one general region to another using the boundary curves?
1.) Given Region: Start with a region D in the xy-plane defined by boundary curves. 2.) Transformation: Use a transformation T to map points from D to a new region R in another coordinate system. 3.) Boundary Curves: Identify the boundary curves of D and express them in terms of the new variables.
How to find surface integrals of vector fields?
1.) Given a Surface and Vector Field: Express the surface with parametric equations r(u,v) and the vector field F. 2.) Surface Integral Formula: For a vector field F=⟨P,Q,R⟩, the surface integral is given by: ∬S F⋅dS=∬D F(r(u,v))⋅(ru×rv)dudv 3.) Evaluate the Integral: Integrate the dot product over the parameter domain D. ∬S F⋅dS=∬D F(r(u,v))⋅(ru×rv)dudv
How to find the surface area with a surface integral?
1.) Given a Surface: Express the surface with parametric equations r(u,v). 2.) Surface Area Formula: The surface area A is given by: A=∬S 1dS=∬D∣ru×rv∣dudv 3.) Evaluate the Integral: Integrate the magnitude of the cross product over the parameter domain D. A=∬S 1dS=∬D ∣ru×rv∣dudv
How to evaluate a surface integral?
1.) Given a Surface: Express the surface with parametric equations r(u,v). 2.) Surface Integral Formula: For a scalar function f(x,y,z), the surface integral is given by: ∬S f dS=∬D f(r(u,v))⋅∣ru×rv∣dudv 3.) Evaluate the Integral: Integrate the function f over the parameter domain D. ∬S f dS=∬D f(r(u,v))⋅∣ru×rv∣dudv
How to use Green's Theorem to find Work?
1.) Given: Start with a vector field F=⟨P,Q⟩ and a curve C parametrized by r(t)=⟨x(t),y(t)⟩. 2.) Line Integral: The work done by F along C is given by: Work=∮C *F*⋅dr 3.) Green's Theorem: Use Green's Theorem to express the line integral as a double integral over the region D enclosed by C: Work=∬D (∂Q/∂x−∂P/∂y) dA
How to draw a vector field?
1.) Grid: Create a grid of points in the xy-plane. 2.) Vectors: At each point, draw a vector based on the values of the vector field at that point. 3.) Visualize: Use arrows to represent the direction and length of the vectors.
What is the Jacobian of T and how to use it?
1.) Jacobian Determinant ∣JT∣: The Jacobian determinant of a transformation T(x,y) is given by: ∣JT∣=∣∣∂(x,y)∂(u,v)∣∣ 2.) Use in Integration: When applying a change of variable in integration, the Jacobian determinant scales the function and the differential area or volume element. ∬D f(x,y) dA= ∬R f(u,v)⋅∣JT∣dudv ∭E g(x,y,z) dV= ∭W g(u,v,w)⋅∣JT∣dudvdw
How to parametrize a curve in order to evaluate the line integrals?
1.) Parameterization: Express the curve C as a vector-valued function r(t)=⟨x(t),y(t),z(t)⟩ with a≤t≤b. 2.) Limits: Determine the limits of integration by the parameter t. 3.) Differential Arc Length: For line integrals with respect to arc length, use ds=√((dtdx)^2+(dtdy)^2+(dtdz)^2)dt. 4.) Substitute: Substitute the parametric equations and ds into the line integral. ∫C f(x,y,z) ds= ∫ab f(r(t))⋅∣r′(t)∣dt
How to evaluate line integrals both in space and in vectors?
1.) Parameterization: Parametrize the curve C using r(t). 2.) Calculate Differential: Compute the differential element (dx, dy, ds) depending on the type of line integral. 3.) Substitute: Substitute the parametric equations and the differential element into the line integral. 4.) Integrate: Evaluate the resulting integral. ∫C f(x,y,z) ds=∫ab f(r(t))⋅∣r′(t)∣dt ∫C *F*⋅d*r*=∫ab *F*(*r*(t))⋅*r*′(t)dt
What are vector fields both algebraically and visually?
Algebraically: A vector field is a function that assigns a vector to each point in space. Mathematically, F(x,y,z)=⟨P(x,y,z),Q(x,y,z),R(x,y,z)⟩. Visually: In a vector field, vectors at each point indicate the direction and magnitude of a physical quantity.
How does curl relate to a conservative vector field?
Curl and Conservative Vector Field: - If F is a conservative vector field (F=∇f), then ∇×F=0. - Conversely, if ∇×F=0 in a simply connected region, then F is conservative. ∇×F=0⟹F is conservative
How does the Curl relate to Green's Theorem?
Curl and Green's Theorem: - For a vector field *F*=⟨P,Q⟩ on a simply connected region D with a positively oriented curve C:
What does it mean if the Curl is 0?
Curl is 0 (∇×F=0): - Indicates that the vector field F has no rotation or "twirl" at each point. Implies that F could be a conservative vector field. ∇×F=0⟹No rotation or "twirl"
What is Curl and Divergence, what does it tell us, how useful is it, and how to know if the result is a vector or a scalar?
Curl ∇×F: - Definition: Measures the rotation or "twirl" of a vector field F. - Formula: Curl F = ∇×F = (∂R/∂y -∂Q/∂z)*i*+(∂P/∂z -∂R/∂x)*j*+(∂Q/∂x -∂P/∂y)*k* Divergence ∇⋅F: - Definition: Measures the rate at which a vector field F spreads or diverges from a point. - Formula: ∇⋅F = ∂P/∂x+∂Q/∂y+∂R/∂z - Usefulness: Curl and divergence provide insight into the behavior of vector fields. - Result Type: Curl gives a vector, and divergence gives a scalar.
What are cylindrical coordinates and how do you plot points in cylindrical coordinates? Know what shapes are easy to describe with cylindrical coordinates.
Cylindrical coordinates (r,θ,z) represent a point in space. To plot points: x=rcos(θ) y=rsin(θ) z=z Shapes that are easy to describe in cylindrical coordinates include cylinders and objects with rotational symmetry around the z-axis.
What are Line integrals, what do they tell us and how are they different from the other integrals you have seen?
Definition: A line integral is an integral along a curve C that measures the total effect of a vector field along that curve. Meaning: It tells us how a scalar or vector quantity is distributed along a curve in a vector field. Difference: Unlike definite integrals over intervals, line integrals are over curves, and the result depends on the path taken. ∫C f(x,y,z) ds
What does it mean for a vector field to be conservative?
Definition: A vector field F(x,y,z)=⟨P(x,y,z),Q(x,y,z),R(x,y,z)⟩ is conservative if it is the gradient of a scalar function f(x,y,z). Implication: If *F*=∇f, then ∮C *F*⋅dr=f(r₂)−f(r₁) for any curve C from r₁ to r₂. Path Independence: The line integral of a conservative field between two points is independent of the path taken.
What does it mean if the Divergence is 0?
Divergence is 0 (∇⋅F=0): - Indicates that the vector field F has no "source" or "sink" at each point. - Implies that F could be an incompressible vector field. ∇⋅F=0⟹No "source" or "sink"
For the solids you are integrating over, how do you draw the solid along with the corresponding traces? Be able to express the solid in set notation and determine the bounds of the solid.
Draw Solid and Traces: - For Type I, draw traces on the yz-plane. - For Type II, draw traces on the xz-plane. - For Type III, draw traces on the xy-plane. Express Solid in Set Notation: - For Type I, E={(x,y,z)∣a≤x≤b,g1(y,z)≤y≤g2(y,z),c≤z≤d} - For Type II, E={(x,y,z)∣a≤x≤b,h1(x,z)≤y≤h2(x,z),c≤z≤d} - For Type III, E={(x,y,z)∣k1(x,y)≤z≤k2(x,y),a≤x≤b,c≤y≤d} Determine Bounds: Use the set notation to traces to find the bounds for x,y,z accordingly. Order of Integration: Determine the order by examining the bounds and choosing the most convenient one.
How to apply Green's Theorem where D is a simple region and when D is a finite union of simple regions?
For a Simple Region D: 1.) Identify D and its boundary C. 2.) Express the vector field F=⟨P,Q⟩. 3.) Verify that the conditions for Green's Theorem are satisfied. 4.) Apply Green's Theorem: ∮C *F*⋅dr=∬D (∂Q/∂x−∂P/∂y) dA For D as a Finite Union of Simple Regions: 1.) Decompose D into simpler regions, each satisfying the conditions. 2.) Apply Green's Theorem separately to each region. 3.) Sum up the results to obtain the total circulation or flux. ∮C *F*⋅dr=∬D (∂Q/∂x−∂P/∂x) dA
How to integrate over a rectangular region or a more generic shape?
For a rectangular region, the bounds can be given as ∫ab∫cd f(x,y) dydx or ∫cd∫ab f(x,y) dxdy. (if constants, then flip order) For a more generic shape, identify the appropriate bounds based on the geometry of the region (Image).
How to be able to express and draw a region in both Rectangular and Polar Coordinates?
For rectangular coordinates, express the region using inequalities for x and y. For polar coordinates, express the region using inequalities for r and θ. To draw the region, use the corresponding coordinate axes, highlighting the specified area.
What is the Fundamental Theorem of Line Integrals, what does it tell us, and how is it useful?
Fundamental Theorem: If F is a conservative vector field on a simply connected region D and *F*=∇f, then for any curve C from r₁ to r₂ in D: ∫C *F*⋅dr=f(r₂)−f(r₁) Meaning: It relates the line integral of a conservative vector field to the values of a potential function f at the endpoints of the curve. Usefulness: Simplifies the computation of certain line integrals and provides a method for finding potential functions.
What is Green's Theorem, what does it tell us, how is it useful, when to apply it, and when does it work. Conditions are important.
Green's Theorem: For a vector field F=⟨P,Q⟩ on an open region D whose boundary is a piecewise smooth, positively oriented curve C: ∮C *F*⋅dr=∬D (∂Q/∂x−∂P/∂y) dA Meaning: Relates a line integral around a closed curve to a double integral over the region enclosed by the curve. Usefulness: Simplifies line integrals into double integrals and provides a way to evaluate circulation and flux. Conditions for Green's Theorem: D must be a simply connected region, and ∂P/∂y and ∂Q/∂x are cont. on D.
In a cylindrical coordinate system, what does the region of integration look like?
In cylindrical coordinates, the region of integration often looks like a cylinder or a portion of a cylinder. The bounds for r and θ define the circular base, and the bounds for z determine the height of the region.
In a spherical coordinate system, what does the region of integration look like?
In spherical coordinates, the region of integration often looks like a portion of a sphere. The bounds for ρ determine the radius, θ determines the azimuthal angle, and ϕ determines the polar angle.
What does it mean for a line integral to be independent of path and how can we determine if a line integral is independent of path?
Independence of Path: A line integral is independent of path if the value of the integral is the same for any two paths with the same initial and terminal points. Determining Independence: If ∫C *F*⋅dr is independent of path, then *F* is conservative. Alternatively, if the line integral of F over any closed curve is zero, then F is conservative. ∮C *F*⋅dr=0 ⟹ *F* is conservative
What's the difference between a line integral of a function with respect to x, with respect to y, and with respect to arc length? Be comfortable with the notation that goes along with each of these.
Line Integral with Respect to x: Denoted as ∫C /M/ dx, it measures the work done along the curve C in the x-direction. Line Integral with Respect to y: Denoted as ∫C /N/ dy, it measures the work done along the curve C in the y-direction. Line Integral with Respect to Arc Length (s): Denoted as ∫C /f/ ds, it measures the work done along the curve C with respect to the arc length s. ∫C P dx + Q dy=∫C *F*⋅dr
How to find moments, center of mass, and moments of inertia?
Moments (Mx,My,): Mx=∬D xρ(x,y,z) dV My=∬D yρ(x,y,z) dV Center of Mass (xˉ,yˉ): xˉ=Mx/m xˉ=(1/m)∬D xρ(x,y,z) dV yˉ=My/m yˉ=(1/m)∬D yρ(x,y,z) dV Moments of Inertia (Ix,Iy,Iz): Ix=∬D y^2ρ(x,y,z)dV (inertia about the x-axis) Iy=∬D x^2ρ(x,y,z)dV (inertia about the y-axis)
What is an oriented surface. Positive orientation vs negative orientation?
Oriented Surface: - Definition: An oriented surface is a surface with a chosen direction or orientation. - Positive Orientation: The orientation that follows the right-hand rule for the parametric equations. - Negative Orientation: The opposite orientation to the positive orientation. Importance: Orientation affects the direction of normal vectors and the sign of surface integrals.
How to be able to express a surface with parametric equations?
Parametric Equations for a Surface: - Express a surface using parametric equations r(u,v)=⟨x(u,v),y(u,v),z(u,v)⟩. Parameters u & v: - These are variables that parameterize the surface, defining points on the surface as functions of u & v.
How to describe the movement of an object if given a vector field?
Particle Path: If a particle moves according to a vector field F(x,y,z), its path is described by solving the system of differential equations: dx/dt=P(x,y,z) dy/dt=Q(x,y,z) dz/dt=R(x,y,z) Trajectory: The trajectory of the particle is the curve traced out in space.
How can we determine if a vector field is conservative?
Path Independence: If the line integral of F over any closed curve is zero, then F is conservative. Theorem: If F(x,y)=P(x,y)/i/ +Q(x,y)/j/ is a conservative vector field, where P and Q have cont. first order derivatives on domain D, then throughout D we have ∂P/∂y = ∂Q/∂x,
What is the potential of a function and how to find it?
Potential Function: If F(x,y,z)=⟨P(x,y,z),Q(x,y,z),R(x,y,z)⟩ is conservative, it has a potential function f(x,y,z) such that F=∇f. Finding the Potential: To find f, integrate each component of F with respect to the corresponding variable. f(x,y,z)=∫ P dx+∫ Q dy+∫ R dz. Checking Conservativeness: Verify that the partial derivatives of f match the components of F. F=∇f
What are the properties of integration and how to use them?
Properties of integration include linearity, additivity, and the constant multiple rule. Linearity allows you to factor constants out of integrals Additivity allows you to split integrals over the sum of functions Constant multiple rule allows you to factor constants out of the integrand. Use these properties to simplify integrals.
What is the relationship between curl and divergence?
Relationship Between Curl and Divergence: For a vector field F=⟨P,Q,R⟩: ∇⋅(∇×F)=0 ∇×(∇⋅F)=0 There is an "orthogonality" between the operations.
What is the relationship between surface integrals and line integrals?
Relationship Between Surface and Line Integrals: - A surface integral over a vector field F is analogous to a line integral over the same vector field along the boundary of the surface. - Green's Theorem relates a line integral to a double integral and is a special case of the surface integral. ∬S *F*⋅dS = ∮C *F*⋅dr
Coming Soon...Stokes Theorem
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What are spherical coordinates and how do you plot points in spherical coordinates? Know what shapes are easy to describe with spherical coordinates.
Spherical coordinates (ρ,θ,ϕ) represent a point in space. To plot points: x=ρsin(ϕ)cos(θ) y=ρsin(ϕ)sin(θ) z=ρcos(ϕ) Shapes that are easy to describe in spherical coordinates include spheres and objects with spherical symmetry.
What is the definition of a surface integral?
Surface Integral: - Definition: The surface integral of a function f(x,y,z) over a surface S is denoted as ∬S f dS. - Formula: ∬S f dS=∬D f(r(u,v))⋅∣ru×rv∣dudv. - Represents: The flux of f across the surface S. ∬S f dS=∬D f(r(u,v))⋅∣ru×rv∣dudv
What is the basic algebraic evaluation of a double integral?
The basic algebraic evaluation of a double integral is done by iteratively integrating the function over each variable, following these steps: 1.) Integrate with respect to the inner variable. 2.) Treat the outer variable as a constant. 3.) Integrate the result from step 1 with respect to the outer variable.
What are the definitions of double integrals over rectangles in terms of Riemann sums? What does the double integral represent? How to graphically show what the double integral represents.
The double integral of a function over a rectangle is defined using Riemann sums, where you sum up the function values times the area of small rectangles within the region. The double integral represents the volume under the surface of a function over a region in the xy-plane. Graphically, the double integral represents the accumulation of volumes of infinitesimally small rectangular columns stacked over the region.
How to use density to find the mass of an object, being comfortable writing the equation for density if given information about the density being proportional to another value?
The mass (m) of an object with density (ρ) can be found using the triple integral: m=∭E ρ(x,y,z) dV If density is proportional to another value, say k, then ρ(x,y,z)=k⋅g(x,y,z), where g(x,y,z) is the given information.
Where does the surface area equation come from? How to find the surface area?
The surface area (S) is calculated using the double integral: S=∬R √(1+(∂f/∂x)^2 + (∂f/∂y)^2) dA. Where R is the region in the xy-plane bounded by the curve C defined by f(x,y)=0 Steps to find surface area: 1.) Express the surface in the form z=f(x,y). 2.) Compute the partial derivatives ∂f/∂x & ∂f/∂y. 3.) Plug the derivatives into the surface area formula and integrate over the region R. The formula comes from approximating the surface by tiny rectangles and summing up their areas, then taking the limit as the rectangles become infinitesimally small. Read questions carefully. Often students will not use the correct integrand b/c they do not realize they are asked for surface area and not volume.
What is the definition of triple integrals and how do they relate to Riemann sums?
The triple integral of a function f(x,y,z) over a region E is defined as: ∭E /f/(x,y,z) dV=limΔV→0∑∑∑f(xi,yj,zk)⋅ΔVijk Where ΔVijk is the volume of a small rectangular box within the region, and the limits of summation approach infinity as the volume of the boxes approaches zero.
How to approximate double integrals using Riemann sums and the Midpoint Rule if given an equation, a table, or a contour plot.
To approximate double integrals using Riemann sums, divide the region into small rectangles, evaluate the function at the midpoint of each rectangle, and sum up the values times the area. The Midpoint Rule involves evaluating the function at the midpoint of each rectangle and using those values in the Riemann sum.
How to be able to change an integral from Rectangular to Polar Coordinates. Understand where the formula comes from and in a polar coordinate system, what does the region of integration look like?
To change from rectangular to polar coordinates: ∬R f(x,y) dA=∫θ₁θ₂∫r₁(θ)r₂(θ) f(rcos(θ),rsin(θ))⋅rdrdθ Where r₁(θ) & r₂(θ) are the radial bounds and θ₁ & θ₂ are the angular bounds. In polar coordinates, the region of integration looks like a wedge or a sector, depending on the bounds.
How to change over to Cylindrical Coordinates and know why the equation is what it is?
To change to cylindrical coordinates: x=rcos(θ) y=rsin(θ) z=z The equation comes from the relationships between rectangular and cylindrical coordinates. These equations express x & y in terms of r & θ, providing an alternative description of a point in space.
How to change over to Spherical Coordinates and know why the equation is what it is?
To change to spherical coordinates: x=ρsin(ϕ)cos(θ) y=ρsin(ϕ)sin(θ) z=ρcos(ϕ) The equation comes from the relationships between rectangular and spherical coordinates. These equations express x, y, and z in terms of ρ, θ, and ϕ, providing an alternative description of a point in space.
How to be able to convert between Rectangular and Polar Coordinates?
To convert from rectangular to polar coordinates: x=rcos(θ), y=rsin(θ), x^2+y^2=r^2 To convert from polar to rectangular coordinates: r^2=x^2+y^2, tan⁻¹(y/x)=θ
How to evaluate an integral in Polar coordinates over general regions?
To evaluate a double integral in polar coordinates over a general region D: 1.) Express the function f(x,y) in terms of r & θ. 2.) Determine the bounds for r & θ by understanding the shape of D. 3.) Rewrite the integral in polar form. 4.) Integrate with respect to r & then θ, respecting the bounds. ∬D f(x,y) dA=∫αβ∫ab f(rcos(θ),rsin(θ))⋅rdrdθ
How to find the average value.
To find the average value of a function over a region, calculate the double integral of the function over the region and then divide by the area of the region. Symbolically, the average value *(favg) is given by: favg = 1/Area(D)∬D f(x,y) dA* The average value of f over interval [a,b] is the image
How to use the double integral to find the volume and area?
To find the volume, the double integral is expressed as ∬D f(x,y) dA, where f(x,y) represents the height of the surface. To find the area, use ∬D 1 dA or ∬D √(1+(∂f/∂x)^2 + (∂f/∂y)^2) dA.
How to rewrite an integral if the order of integration is switched?
To switch the order of integration in a triple integral, follow these steps: 1.) Identify the region of integration in both orders. 2.) Express the integrand in terms of the other variables. 3.) Reverse the order of integration. ∭E f(x,y,z)dV=∫ab∫cd∫p(x,y)q(x,y) f(x,y,z) dzdydx
How to be able to switch the order of integration when necessary?
To switch the order of integration: 1.) Identify the region of integration. 2.) Determine the bounds for the opposite order by projecting the region onto the corresponding axis. 3.) Integrate with the new order. If the original bounds are a≤x≤b and c≤y≤d, switching the order would mean integrating with respect to y first and then with respect to x or vice versa.
How to recognize when to use Type I and Type II Region? Be able to draw the surface over the given region. Be able to draw the region. Be able to identify the bounds and order of integration.
Type I Region: When the region is between two curves x = g₁(y) and x = g₂(y), choose the order dydx. Type II Region: When the region is between two curves y = h₁(x) and y = h₂(x), choose the order dxdy. Draw the region, identify the curves, and set up the bounds accordingly.
How to evaluate triple integrals over general regions (remember there are three types)?
Type I Region: ∭E f(x,y,z) dV =∫ab∫g₁(x)g₂(x)∫h₁(x,y)h₂(x,y) f(x,y,z) dzdydx Type II Region: ∭E f(x,y,z) dV =∫cd∫h₁(y)h₂(y)∫g₁(x,y)g₂(x,y) f(x,y,z) dzdxdy Type III Region: ∭E f(x,y,z) dV =∫g₁(z)g₂(z)∫h₁(z)h₂(z)∫f₁(x,y,z)f₂(x,y,z) f(x,y,z) dxdydz
How to find volume, mass, moments, and center of mass using applications of triple integrals?
Volume (V): V=∭E 1 dV Mass (m): m=∭E ρ(x,y,z) dV Moments about coordinate planes (Myz,Mxz,Mxy): Myz=∭E x ρ(x,y,z) dV Mxz=∭E y ρ(x,y,z) dV Mxy=∭E z ρ(x,y,z) dV Center of Mass (xˉ,yˉ,zˉ): xˉ=Myz/m yˉ=Myxz/m zˉ=Mxy/m Moments of Inertia (Ix,Iy,Iz): Ix=∭E (y2+z2) ρ(x,y,z) dV Iy=∭E (x2+z2) ρ(x,y,z) dV Iz=∭E (x2+y2) ρ(x,y,z) dV
How and when to change an integral from Rectangular to Cylindrical Coordinates?
When: Change to cylindrical coordinates when the region D is more naturally described or bounded in cylindrical coordinates, often when there is z-axis symmetry. Steps: 1.) Express the function f(x,y,z) in terms of r,θ,z. 2.) Replace x & y with rcos(θ) & rsin(θ), respectively. 3.) Replace dA with rdrdθ. 4.) Replace dz with dz. ∭E f(x,y,z) dV=∭E f(rcos(θ),rsin(θ),z)⋅rdrdθdz
How and when to change an integral from Rectangular to Spherical Coordinates?
When: Change to spherical coordinates when the region D is more naturally described or bounded in spherical coordinates, often when there is symmetry with respect to the origin. Steps: 1.) Express the function f(x,y,z) in terms of ρ,θ,ϕ. 2.) Replace x, y, and z with the spherical coordinate equations. 3.) Replace dV with ρ2sin(ϕ)dρdθdϕ. ∭E f(x,y,z) dV=∭E f(ρsin(ϕ)cos(θ),ρsin(ϕ)sin(θ),ρcos(ϕ))⋅ρ2sin(ϕ)dρdθdϕ
