Exam 2 (This One)

Question 1

Trig Sub: sqrt{x^2+a^2}

Answer 1

x = a*tan(theta)

Question 2

Trig Sub: sqrt(x^2 - a^2)

Answer 2

x = a*sec(theta)

Question 3

Trig Sub: sqrt(a^2 - x^2)

Answer 3

x = a*sin(theta)

Question 4

Partial Fraction Decomposition: (x^2 - 7x + 12)/((x - 1)(x+2)^2(x^2+x-1)(x^2 + 4)^2)

Answer 4

A/(x-1) + B/(x+2) + C/(x+2)^2 + (Dx + E)/(x^2 + x -1) + (Fx + G)/(x^2 + 4) + (Hx + I)/(x^2 + 4)^2

Question 5

Partial Fraction Decomposition: Integrate(A/(mx + b))

Answer 5

ln(u)

Question 6

Partial Fraction Decomposition: Integrate(A/(mx + b)^n

Answer 6

Simple u substitution -> Integrate(u^-n)

Question 7

Partial Fraction Decomposition: Integrate(A/(x^2 + b^2))

Answer 7

Arctangent

Question 8

Partial Fraction Decomposition: Integrate(A/(ax^2 + bx + c))

Answer 8

Complete the Square on Bottom

Question 9

Partial Fraction Decomposition: Integrate((Ax + B)/(ax^2 + bx + c))

Answer 9

Break into Two Integrals

Question 10

Arc Length

Answer 10

Integrate[a, b] sqrt(1+f’(x)^2)

Question 11

Surface Area (Rotate about x)

Answer 11

2pi*Integrate[a, b] f(x) sqrt(1+f’(x)^2)

Question 12

Surface Area (Rotate about y)

Answer 12

2pi*Integrate[a, b] x sqrt(1+f’(x)^2)

Question 13

Surface Area (Rotate about x = d)

Answer 13

2pi*Integrate[a, b] |x - d| sqrt(1 + f’(x)^2)

Question 14

Integrate[a, b] wD(h)L(h)dh

Answer 14

D(h) is the depth at h
L(h) is the width of the plate at h
w is the weight density of water

Question 15

Centroid: Moment about the y axis

Answer 15

Integrate[a, b] x (f(x) - g(x))dx

Question 16

Centroid: Moment about the x axis

Answer 16

Integrate[a, b] 1/2 (f^2(x) - g^2(x))dx

Question 17

Area of a Centroid

Answer 17

Integrate[a, b] (f(x) - g(x)) dx

Question 18

Center of mass for Centroid

Answer 18

Xbar = M_y/A
Ybar = M_x/A