Area = ∫[Height] Width
A = ∫(f(x) - g(x)] dx
1. Graph Equasions
2. Label
3. Determine how to slice
4. Set up dA
5. dA = height*dx
6. Get range a & b from intersections
7. Plug in and find area

Volume by Disk

dV = A(x) dx
V = ∫A(x)dx
Volume = ∫Radius2 * Thickness
V = ∫(pi(r)2) dx

Volume by Washer

dV = A(x) dx
V = ∫A(x) dx
Volume = ∫[(pi r out2)-(pi r in2)] dx

Slice Perpendicular to Axis of Rotation

Volume by Shell

dVolume = Circumference * dArea
dV = (2 pi r) dArea
V = ∫(2 pi r)(Area)dx
1. Write: dV = 2 pi r dA
2. Find dA(height dx)
3. Find Radius(x or y)
4. Plug in
5. Take integral

Slice Parallel to Axis of Rotation

Average Value of a Function

Average Value = 1/b-a * ∫f(x) dx
Symmetry:
If f(x) is EVEN, then ∫f(x)dx from -a to a = 2∫f(x) from 0 to a
If f(x) is ODD, then ∫f(x)dx from -a to a = 0

Important Integrals

∫c f(x) dx

= c ∫f(x) dx

∫[f(x) + g(x)] dx

= ∫f(x) dx + ∫g(x) dx

∫ 1/x dx

= ln|x| + C

∫ex dx

= ex + C

∫bx dx

= (bx / lnb) + C

Methods of Integration

Method

When to Use

Example

U-Substitution

When a Polynomial is raised to a power > 1

∫(3x + 5)5

Integration by Parts

When U-Sub will not work

∫xex

Trigonometric Integration

Only Trig raised to powers

∫sin6x cos3xdx

Trigonometric Substitution

3/2 powers or Sqrt(a2-x2) etc.

dx/(x2Sqrt(25-x2))

Integration by Parts

Logarithmic
Inverse trig
Algebraic
Trigonometric
Exponential
∫u dv = u v - ∫v du
1. Write u v - ∫v du
2. Use LIATE to find u; the other term becomes dV
3. Setup u= dV= du= V=
4. Solve

Cyclical Functions will need to be split and substituted.

Trigonometric Integration

Identities
sin2t+cos2t = 1
sin2t = 1/2 [1-cos(2t)]
cos2t = 1/2 [1+cos(2t)]
Can use with U-Substitution

Don’t change all of the trig to the same form.

Trigonometric Integration

Identities
sin2t+cos2t = 1
sin2t = 1/2 [1-cos(2t)]
cos2t = 1/2 [1+cos(2t)]
Can use with U-Substitution

Don’t change all of the trig to the same form.

Trigonometric Substitution

Pythag. Identities
sin2 + cos2 = 1
1 + tan2 = sec2
1 + cot2 = csc2
1. Identify a and u
2. Sub in the trig
3. Manipulate to simplify
4. Get rid of trig with a triangle

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