Exam 1

Question 1

Volume of Revolution: If the axis of rotation is perpendicular to the axis of integration, then you want to use which method?

Answer 1

Shell Method

V = [a, b] (2pi * (Radius)(Height))

Question 2

What do you use when you don’t have a volume of revolution? (Perpendicular to the axis of integration)

Answer 2

Slicing

V = Integrate [a, b] A(x)

Question 3

Work =

Answer 3

Force * Displacement

integrate [a, b] F(x)

Question 4

Work: If D(h) represents the height travelled at depth h, and A(h) is the cross section area, where w is the weight density, the work becomes:

Answer 4

integrate [a, b] (wD(h)A(h)) dh

Question 5

Average Value of a Function

Answer 5

f(c) = (1/(b-a)) * integrate [a, b] f(x) dx

a) Find the avg value of the function(s)
b) Find a c in the interval on which the function achieves its avg value

Question 6

Integration by Parts

Answer 6

Integrate udv = uv - integrate (vdu)

Question 7

Trig Integrals:
Integrate ((sin^m)x)((cos^n)x)dx
If n is odd:

Answer 7

Save one cosx and convert the rest to sin using

(cos^2)x) = 1 - ((sin^2)x

Question 8

Trig Integrals:
Integrate ((sin^m)x)((cos^n)x)dx
If m is odd:

Answer 8

Save one sinx and convert the rest to cos using

(sin^2)x) = 1 - ((cos^2)x

Question 9

Trig Integrals:
Integrate ((sin^m)x)((cos^n)x)dx
If both m and n are even:

Answer 9

Use the identites:
((sin^2)x) = ((1-cos2x)/2)
((cos^2)x) = ((1+cos2x)/2)

Question 10

Trig Integrals:
Integrate ((tan^m)x)((sec^n)x)dx
If n is even

Answer 10

Save on ((sec^2)x) and convert the rest to tan using
((sec^2)x) = ((tan^2)x) + 1

Question 11

Trig Integrals:
Integrate ((tan^m)x)((sec^n)x)dx
If m is odd

Answer 11

Save a secxtanx and convert the rest to sec using

((tan^2)x) = ((sec^2)x) - 1

Question 12

Trig Integrals:
Integrate ((tan^m)x)((sec^n)x)dx
If m is even and n is odd

Answer 12

Convert everything to sec and integrate by parts with

dv = ((sec^2)x)

Question 13

Trig Integrals:

integrate ((tan^n)x)dx

Answer 13

convert one ((tan^2)x) to ((sec^2)x) -1 
split the problem into two integrals

Question 14

Trig Integrals:

Integrate secx dx

Answer 14

= ln |secx + tanx| + C

Question 15

Trig Substitution:

a^2 - (b^2)(x^2)

Answer 15

x = (a/b)sin(theta)

Question 16

Trig Substitution:

(b^2)(x^2) + a^2

Answer 16

x = (a/b)tan(theta)

Question 17

Trig Substitution:

(b^2)(x^2) - a^2

Answer 17

x = (a/b)sec(theta)

Question 18

Trig Substitution:

If the integrand involves ax^2 + bx + c:

Answer 18

complete the square to get it into the form a(x-h)^2 + k.
After factoring out the a and applying the substitution
u = x - h, the integrand will then fit one of the three forms

Question 19

Volume of Rotation: If the axis of rotation is parallel to the axis of integration (horizontal line, integrating in x, or vertical line, integrating in y) Then you want to use which method.

Answer 19

Washer Method

V = integrate [a, b] (pi * (R^2 - r^2))