LA Tech Barron MATH243 - Exam Three

Match the integrals with the type of coordinates which make them the easiest to do. Put the letter of the coordinate system to the left of the number of the integral. 1. ∫∫∫ z dV where E is: 1≤x≤2, 3≤y≤4, 5≤z≤6 2. ∫∫∫ z^2 dV where E is: −2≤z≤2, 1≤x2+y2≤2 3. ∫∫ 1/x dx dy 4. ∫∫ 1/(x^2+y^2) dA where D is: x2+y2≤4 5. ∫∫∫E dV where E is: x^2+y^2+z^2≤4, x≥0, y≥0, z≥0 A. polar coordinates B. spherical coordinates C. Cartesian coordinates D. cylindrical coordinates

15.8 "spherical coordinates"

What are the cylindrical coordinates of the point whose spherical coordinates are (5, −3, (5π)/6) ? r = θ = z=

15.8 "spherical coordinates"

What are the rectangular coordinates of the point whose spherical coordinates are (2, (1/2)π, −(1/6)π) ? x= y= z=

15.8 "spherical coordinates"

What are the spherical coordinates of the point whose rectangular coordinates are (2, 4, 5) ? ρ = θ = Φ =

15.8 "spherical coordinates"

Use spherical coordinates to evaluate the triple integral ∭(e^-(x^2+y^2+z^2)) / (√(x^2+y^2+z^2)) dV where E is the region bounded by the spheres x^2+y^2+z^2=4 and x^2+y^2+z^2=9.

15.8 "spherical coordinates" "spheres"

Find the volume of the ellipsoid x^2+y^2+5 z^2=25.

15.8 "volume...ellipsoid"

Find the velocity and position vectors of a particle with acceleration a(t)=⟨0,0,1⟩, and initial conditions v(0)=⟨−5,0,−5⟩ and r(0)=⟨5,3,−1⟩. v(t)=⟨_____, _____, _____⟩ r(t)=⟨_____, _____, _____⟩

13.4 "velocity and position vectors" "acceleration"

If r(t)=cos(10t) i + sin(10t) j −7t k, compute: A. The velocity vector v(t)= _____ i+ _____ j+ _____ k B. The acceleration vector a(t)= _____ i+ _____ j+ _____ k

13.4 "velocity vector" "acceleration vector"

Find the velocity, acceleration, and speed of a particle with position function r(t)=⟨3t sin(t), 3t cos(t), −6t^2⟩ v(t)=⟨_____, _____, _____⟩ a(t)=⟨_____, _____, _____⟩ |v(t)|=_____

13.4 "velocity, acceleration, and speed"

Find the velocity, acceleration, and speed of a particle with position function r(t)=⟨−2 cos(t), −8t,−2 sin(t)⟩ v(t)=⟨_____, _____, _____⟩ a(t)=⟨_____, _____, _____⟩ |v(t)|=_____

13.4 "velocity, acceleration, and speed"

Evaluate ∭z(e^(x+y)) dV where B is the box determined by 0≤x≤5, 0≤y≤1, and 0≤z≤3

15.6 "Evaluate ∭z(e^(x+y))dV"

Use a triple integral to find the volume of the solid enclosed by the paraboloid x=y^2+z^2 and the plane x=1.

15.6 "triple integral to find the volume" "paraboloid"

Evaluate the triple integral ∫∫∫xyz dV where E is the solid: 0≤z≤3, 0≤y≤z, 0≤x≤y.

15.6 "triple integral" "∫∫∫xyz dV"

Evaluate the triple integral ∭x^2 e^y dV where E is bounded by the parabolic cylinder z=16−y^2 and the planes z=0, x=4, and x=−4.

15.6 "triple integral" "∭x^2 e^y dV" "parabolic cylinder"

Evaluate the triple integral ∭z dV where E is the solid bounded by the cylinder y^2+z^2=144 and the planes x=0, y=2x and z=0 in the first octant.

15.6 "triple integral" "∭z dV" "solid bounded by the cylinder"

Find the volume of the solid enclosed by the paraboloids z=9(x^2+y^2) and z=8−9(x^2+y^2).

15.6 "volume" "paraboloids"

Express the integral ∭f(x,y,z) dV as an iterated integral in six different ways, where E is the solid bounded by z=0,x=0,z=y−6x and y=12.

15.6 "∭f(x,y,z) dV"

Express the integral ∭f(x,y,z) dV as an iterated integral in six different ways, where E is the solid bounded by z=0,z=3y and x^2=16−y.

15.6 "∭f(x,y,z) dV"

Evaluate the integral by changing to cylindrical coordinates. ∫∫∫(x^2+y^2)^(3/2) dzdydx

15.7 "cylindrical coordinates"

Use cylindrical coordinates to find the volume of the solid that lies within both the cylinder x^2+y^2=1 and the sphere x^2+y^2+z^2=4.

15.7 "cylindrical coordinates"

Use cylindrical coordinates to find the volume of the solid that lies within the sphere x^2+y^2+z^2=81 above the xy-plane and outside the cone z=5√(x^2+y^2)

15.7 "cylindrical coordinates"

What are the cylindrical coordinates of the point whose rectangular coordinates are (x=4, y=5, z=3) ? r= θ= z=

15.7 "cylindrical coordinates"

What are the rectangular coordinates of the point whose cylindrical coordinates are (r=0, θ=2.2, z=10) ? x = y = z =

15.7 "cylindrical coordinates"

A volcano fills the volume between the graphs z=0 and z=1/(x^2+y^2)^(21) and outside the cylinder x^2+y^2=1. Find the volume of this volcano

15.7 "outside the cylinder" "find the volume" "volume between the graphs"

Use cylindrical coordinates to evaluate the triple integral ∫∫∫ √(x^2+y^2) dV where E is the solid bounded by the circular paraboloid z=9−9(x^2+y^2) and the xy-plane.

15.7 "triple integral ∫∫∫ √(x^2+y^2) dV" "cylindrical coordinates"

Use cylindrical coordinates to evaluate the triple integral ∫∫∫x^2 dV where E is the solid that lies within the cylinder x^2+y^2=9, above the plane z=0, and below the cone z^2=25 x^2 + 25 y^2.

15.7 "triple integral ∫∫∫x^2 dV" "cylindrical coordinates"

Find the volume of the ellipsoid x^2+y^2+5 z^2=100

15.7 "volume...ellipsoid"

Find the volume of the solid that lies within the sphere: x^2+y^2+z^2=25 above the xy plane and outside the cone: z=3√(x^2+y^2)

15.8 "sphere"

Match the given equation with the verbal description of the surface: A. Sphere B. Plane C. Elliptic or Circular Paraboloid D. Cone E. Circular Cylinder F. Half plane 1. r^2+z^2=16 2. ρcos(Φ)=4 3. Φ=π/3 4. r=4 5. r=2 cos(θ) 6. ρ=2 cos(Φ) 7. z=r^2 8. ρ=4 9. θ=π/3

15.8 "sphere"

Use spherical coordinates to evaluate the triple integral ∫∫∫x^2+y^2+z^2 dV where E is the ball: x^2+y^2+z^2≤4.

15.8 "spherical coordinates

Evaluate the integral by changing to spherical coordinates. ∫∫∫z√(x^2+y^2+z^2) dz dy dx

15.8 "spherical coordinates"

Assume time t runs from zero to 2π and that the unit circle has been labeled as a clock. Match each of the pairs of parametric equations with the best description of the curve from the following list. Enter the appropriate letter (A, B, C, D, E, F ) in each blank. A. Starts at 12 o'clock and moves clockwise one time around. B. Starts at 6 o'clock and moves clockwise one time around. C. Starts at 3 o'clock and moves clockwise one time around. D. Starts at 9 o'clock and moves counterclockwise one time around. E. Starts at 3 o'clock and moves counterclockwise two times around. F. Starts at 3 o'clock and moves counterclockwise to 9 o'clock.

10.1 "Parametric equations"

Suppose parametric equations for the line segment between (9,−9) and (1,1) have the form: x=a+bt y=c+dt If the parametric curve starts at (9,−9) when t=0 and ends at (1,1) at t=1, then find a, b, c, and d.

10.1 "Parametric equations"

Eliminate the parameter t to find a Cartesian equation for x=−9−t y=−16−4t The Cartesian equation has the form y=mx+b

10.1 "parameter t" + "Cartesian equation"

Assume t is defined for all time. Enter the letter of the graph below which corresponds to the curve traced by the parametric equations. Think about the range of x and y, and whether there is periodicity and or symmetry. 1. x=sin(t)(3−2sin(t));y=cos(t)(3−2sin(t)) 2. x=sin(t);y=cos(t)−2cos(2t) 3. x=t^3/4−t+1;y=t^2/4−1 4. x=sin(t+sin(7t));y=cos(t) 5. x=|cos(t)|⋅cos(t);y=|sin(t)|⋅sin(t)

10.1 "parametric equations"

Write the parametric equations x=√t y=1−t in Cartesian form. y= ___________ with x≥0

10.1 "parametric equations" + "Cartesian form"

Eliminate the parameter t to find a Cartesian equation for: x=t^2 and y=8+3t. and express your equation in the form x=Ay^2+By+C. Then A= B= C=

10.1 "Cartesian equation" + "parameter t"

Find a Cartesian equation relating x and y corresponding to the parametric equations x=e^(3t) y=e^(-6t) Write your answer in the form y=f(x)

10.1 "Cartesian equation" + "parametric equations"

Find the limit: lim(t→0) ⟨ (e^3t−1)/t, t^7/(t^8−t^7), −7/(2+t) ⟩

13.1 "Find the limit"

Match the parametric equations with the graphs labeled A - F. As always, you may click on the thumbnail image to produce a larger image in a new window (sometimes exactly on top of the old one). 1. x=t^2−2, y=t^3, z=t^4+1 2. x=cos(4t), y=t, z=sin(4t) 3. x=cost, y=sin(t), z=ln(t) 4. x=t, y=1/(1+t^2), z=t^2 5. x=sin(3t)*cost,y=sin(3t)*sin(t), z=t 6. x=cos(t), y=sin(t), z=sin(5t)

13.1 "Match the parametric equations"

Find a vector function that represents the curve of intersection of the paraboloid z=3x^2+2y^2 and the cylinder y=3x^2. Use the variable t for the parameter. r(t)=⟨t,____,_____⟩

13.1 "vector function"

Find the unit tangent vector at the indicated point of the vector function r(t)=e^(12t)cos(t) i+e^(12t)sin(t) j+e^(12t) k T(π/2)=⟨_____, ______, ______⟩

13.2

Consider the vector function r(t)=⟨t,t^6,t^3⟩ Compute r′(t)=⟨_____, _____, _____⟩ T(1)=⟨_____, _____, _____⟩ r″(t)=⟨_____, _____, _____⟩ r′(t)×r″(t)=⟨____, ____, _____⟩

13.2 "Compute: r'(t), r"(t)"

Find the derivative of the vector function r(t)=ln(16−t^2) i+√(19+t) j+7e^(−4t) k r′(t)=⟨_____, ______, _____⟩

13.2 "Derivative of the vector function"

Find the derivative of the vector function r(t)=ta×(b+tc), where a=⟨3,5,−2⟩ b=⟨−2,1,−1⟩ c=⟨−5,−5,1⟩. r′(t)=⟨_____, _____, _____⟩

13.2 "Derivative of the vector function"

Find the parametric equations for the tangent line to the curve x=t^2−1, y=t^3+1, z=t^1 at the point (3, 9, 2). Use the variable t for your parameter. x=_____ y=_____ z=_____

13.2 "find...tangent line"

Starting from the point (2,2,3) reparametrize the curve r(t)=(2−3t) i + (2+3t) j + (3+t) k in terms of arclength. r(t(s))= _____ i+ _____ j+ _____ k

13.3 "arclength"

Consider the helix r(t)=(cos(−4t), sin(−4t), −3t). Compute, at t=π/6: A. The unit tangent vector T B. The unit normal vector N C. The unit binormal vector B D. The curvature κ

13.3 "compute...curvature κ"

Find the arclength of the curve r(t)=⟨5t^2, 2√(5) t, ln(t)⟩ 1≤t≤7

13.3 "find the arc length"

Find the arc length of the curve : r(t)=⟨−5 sin(t), 10t,−5 cos(t)⟩ −9≤t≤9

13.3 "find the arclength"

Find the arclength of the curve r(t)=⟨8√2 t, e^(8t), e^(−8t)⟩, 0≤t≤1

13.3 "find the arclength"

Find the curvature κ(t) of the curve r(t)=(−1 sin(t)) i+ (−1 sin(t)) j+(1 cos(t)) k

13.3 "find the curvature κ"

A gun has a muzzle speed of 70 meters per second. What angle of elevation should be used to hit an object 150 meters away? Neglect air resistance and use g=9.8 m/s^2 as the acceleration of gravity.

13.4 "angle of elevation" "speed"

Given that the acceleration vector is a(t)=(−25 cos(5t)) i + (−25 sin(5t)) j + (5t) k, the initial velocity is v(0)=i+k, and the initial position vector is r(0)=i+j+k, compute: A. The velocity vector v(t)= _____ i+ _____ j+ _____ k B. The position vector r(t)= _____ i+ _____ j+ _____ k

13.4 "vector velocity" "position vector" "acceleration vector"