Calc 2 Exam 1 Review - Key Formulas

∫uⁿdu

(uⁿ⁺¹)/(n+1) +C

Integrate ∫sin^mx cosⁿxdx when n is odd and positive

-convert all but one factor of the cosine to sine using cos²x=1-sin²x -expand and integrate

Integrate ∫sin^mx cosⁿxdx when m is odd and positive

-convert all but one factor of the sine to cosine using sin²x=1-cos²x -expand and integrate

Integrate ∫sec^mxtanⁿxdx if there are no secant factors and n is even and positive

-convert one tangent square factor by tan²x=sec²x-1 -expand and repeat if necessary. eventually it will be able to be integrated

Steps for partial fraction decomposition

-divide the rational expression using polynomial long division if it is improper -factor the denominator completely -write the general form of the partial fraction decomposition -multiply both sides of the general form by the LCD the resulting equation is called the basic equation -solve the basic equation for the unknown constants. this can be done by substitution or by coefficient equations

Steps for partial fraction decomposition if a function is improper

-it must be rewritten using polynomial long division and the division algorithm: dividend/divisor = quotient + remainder/divisor -result will be a polynomial and a proper fraction that can be decomposed

Integrate ∫sin^mx cosⁿxdx when both m and n are even

-reduce the powers using the power reduction rules. this may need to be used several times

Integrate ∫sec^mxtanⁿxdx when n is odd and positive

-save a secxtanx factor and convert the rest of the factors to secants using tan²x=sec²x-1 -expand and integrate

Integrate ∫sec^mxtanⁿxdx when m is even and positive

-save a sec²x factor and convert the rest of the factors to tangents using sec²x=1+tan²x -expand and integrate

Integrate ∫sec^mxtanⁿxdx when there are no tangent factors and m is odd and positive

-use integration by parts where u=secx and dv=sec^m-1 x

∫sinudu=

=-cosu+C

∫tanudu=

=-ln|cosu|+C

∫cscudu=

=-ln|cscu-cotu|+C

∫secudu=

=ln|secu+tanu|+C

∫cotudu=

=ln|sinu|+C

∫cosudu=

=sinu+C

Distance between a point (x₁,y₁,z₁) and a plane:

D=|ax₁+by₁+cz₁+d|/√a²+b²+c²

Fluid force

F=w∫h(y)L(y)dy

Torque vector

T=rxF

Trapezoid Rule

TRAPn=∆x/2[f(x₀) + 2f(x₁) + 2f(x₂) +...+ 2f(n-1) +f(xn))

Shell method equation (about y axis)

V=2π∫r(x)h(x)dx

Shell method equation (about x axis)

V=2π∫r(y)h(y)dy

Washer method equation (about x axis)

V=π∫[(f(x))² - (g(x))²]dx

Washer method equation (about y axis)

V=π∫[(f(y))² - (g(y))²]dy

Disk method equation (about x axis)

V=π∫[f(x)]²dx

Disk method equation (about y axis)

V=π∫[f(y)]²dy

Volumes by cross section equation

V=∫A(x)dx

Definition of the displacement vector in physics

W=(|F|cosθ)|D| = |F||D|cosθ = F.D

Work formula

W=∫F(x)dx

Work of a varying force and distance (emptying tanks, hauling chains, etc.)

W=∫f(y)h(y)dy

Hooke's law for springs

W=∫kxdx

Scalar equation of the plane through point (x₀,y₀,z₀) and orthagonal to vector n

a(x-x₀) +b(y-y₀)+c(z-z₀)=0

Dot product of two vectors in 2D

a.b = a₁b₁+a₂b₂

Dot product of two vectors in 3D

a.b =a₁b₁+a₂b₂+a₃b₃

How to calculate the angle between vectors a and b

a.b =|a||b|cosθ

Component form of a vector in 3D

a= <x₂-x₁, y₂-y₁, z₂-z₁>

Component form of a vector in 2D

a= <x₂-x₁, y₂-y₁>

General form of a plane

ax+by+cz+d=0

Cross product (vector product)

axb = <a₂b₃-a₃b₂, a₃b₁-a₁b₃, a₁b₂-a₂b₁> -determinants of matrices multiplied by minor of each element

Scalar projection (component of b along a)

compab = |projab| = (|b|(a.b))/|a||b| = (a.b)/|a|

∫e^u du

e^u +C

Integration by parts for ∫kx√mx+b dx

let u=kx and dv=√mx+b dx. however, this is more easily done by a u-substitution where u = mx+b

Integration by parts for ∫xⁿlnxdx

let u=lnx and dv=xⁿdx. Simplify ∫vdu before integrating. Tabular method ineffective here.

Integration by parts for ∫e^kxsin(mx)dx

let u=sin(mx) and dv=e^kxdx. integrate twice, then collect like terms and solve algebraically

Integration by parts for ∫xⁿsin(kx)dx

let u=xⁿ and dv=sin(kx). works for cos(kx) as well. tabular method works well for power reduction

Integration by parts for ∫xⁿe^kx dx

let u=xⁿ and let dv=e^kxdx. Tabular method works well for power reduction

∫du/u or ∫(1/u)du

ln|u| +C

Vector equations of a plane

n.(r-r₀)=0 or n.r=n.r₀

Equation for a line segment from r₀ to r₁

r(t) = (1-t)r₀ +tr₁ where 0≤t≤1

Arc length formula

s=∫√1+[f'(x)]²dx

Power reduction rules

sin²x= (1-cos(2x))/2 cos²x=(1+cos(2x))/2

Pythagorean Identity

sin²x=1-cos²x cos²x=1-sin²x sec²x=1+tan²x tan²x=sec²x-1

Symmetric equations

x-x₀/a = y-y₀/b = z-z₀/c

Magnitude of torque vector

|T|=|r||F|sinθ

Scalar triple product of three vectors

|a.(bxc)|

Magnitude of the cross product

|axb| =|a||b|sinθ

Magnitude of a vector in 2D

|a|=√a₁²+a₂²

Magnitude of a vector in 3D

|a|=√a₁²+a₂²+a₃²

Length of a vector projection (here, of b onto a)

|projab|=|b|cosθ

Reference angle for a vector

θ=tan⁻¹|y/x|

MRAMn=

∑(f(xk-1)+f(xk))/2 ∆x

RRAMn=

∑f(xk)∆x

LRAMn=

∑f(xk-1)∆x

Area between curves equation

∫f(x)dx - ∫g(x)dx = ∫[f(x)-g(x)]dx

Simpson's Method

∫f(x)dx = ∆x/3[f(x₀) + 4f(x₁)+ 2f(x₂) + 4f(x₃) +...+ 2f(xn-2)+ 4f(xn-1) +f(xn)]

Integration by Parts

∫udv = uv - ∫vdu