Math 240: Calculus III Lab (part II): Week 11 Instructions: Attempt all of these exercises in groups, without using a calculator. If you get stuck on a problem, discuss it with your group. If your entire group is stuck, then your entire group may ask for help. 1.

Evaluate the iterated integral.

π/4 Z 2 ¡

Z 0

2.

0

¢ 3(x tan θ)2 + 3x 2 d x d θ

Evaluate the integral over the rectangular region. ZZ x 2 − y 2 d A, R

1

R = [1, 2] × [0, 1]

3.

Evaluate the iterated integral.

π

Z 6Z 0

4.

0

7r sin2 θ d θ

Evaluate the integral over the rectangular region. ZZ R

7(1 + x 2 ) d A, 1 + y2

R = {(x, y) | 0 ≤ x ≤ 3, 0 ≤ y ≤ 1}

2

5.

Find the volume of the solid in the first octant bounded by the parabolic cylinder z = 25− x 2 and the plane y = 2.

6.

Evaluate the iterated integral. Z 1Z 0

5 1

2xe x dy dx y

3

7.

Roughly sketch the domain D, then evaluate the double integral over D. ZZ 6y d A D = {(x, y) | 0 ≤ x ≤ 1, 0 ≤ y ≤ x 2 } 5 D 4x + 1

8.

Let D be the domain bounded between the curves y = x − 30 and x = y 2 . Set up iterated integrals for both orders of integration, then use the easier one to evaluate the integral. ZZ ydA D

4