math

MATH 162, FALL 2017: ASSIGNMENT 3

Due: Wednesday, November 15, in class

Please answer the following questions, explaining all of the steps that you took to find solutions to the following problems. These problems cover material from Sections 12.1, 12.2, 12.3, 12.6 and 12.8 in the textbook.

1. Evaluate following integrals. a. ∫∫

R

( x

y +

y

x

) dA

where R is the rectangle [1, 2] × [2, 4]. b. ∫ 1

0

∫ x x2

xy + x2 dydx

c. ∫∫∫ E ez/y

where E = {(x, y, z) : 0 ≤ y ≤ 1, y ≤ x ≤ 1, 0 ≤ z ≤ xy}. 2. Let R be the region in the real plane inside of the circle of radius 2

and with positive y-value. a. Express R as a region of type I b. Express R as a region of type II. c. Evaluate the integral∫∫

R xy dxdy

twice; first by viewing it as a region of type I, then by viewing it as a region of type II.

3. a. Find the Jacobian of the variable transformation

x = u + v

2 , y =

u−v 2

.

b. Use part a to compute the double integral∫∫ R

(x−y + 4)3

x + y + 3 dxdy

4. Use polar coordinates to find the volume of the solid bounded by the paraboloids z = 3x2 + 3y2 and z = 4 −y2 −z2.

5. Find the integral∫∫∫ H z3 √

x2 + y2 + z2 dV

MATH 162, FALL 2017: ASSIGNMENT 3

where H is the region of R3 which inside the sphere of radius 2 and center at the origin and above the xy-plane.