Math 241 - Exam #4

Covers 12.1-12.8, 10.5, 11.4

Solutions

1. int_-1^1 int_0^5 x^2 y dy dx = int_-1^1 25/2 x^2 dx = 25/3

2. int_1^3 int_1+x^2x (x-2y) dy dx = int_1^3 -2x^2 + x + 1 dx = -34/3

3. Surface area = int_0^1 int_0^y sqrt(2+4y^2) dx dy
= int_0^1 sqrt(2+4y^2)y dy = 1/12 (2+4y^2)^(3/2) |_0^1
= 1/12(6^(3/2) - 2^(3/2))

4. int_0^1 int_x^2^x int_0^x (x+2y) dz dy dx
= int_0^1 int_x^2^x (x^2 + 2xy) dy dx
= int_0^1 2x^3 - x^4 -x^5 dx = 2/15

5. 1/8th of a sphere, in the first octant (x,y,z>0)
V = pi/6

6. int_0^1 int_0^x^2 sqrt(x^3+1) dy dx

7. int_0^2pi int_2^5 r^2 dr dtheta

8. Five others: 0 < x < 1, 0 < z < 1-sqrt(x), sqrt(x) < y < 1-z
0 < y < 1, 0 < x < y^2, 0 < z < 1-y
0 < y < 1, 0 < z < 1-y, 0 < x < y^2
0 < z < 1, 0 < y < 1-z, 0 < x < y^2
0 < z < 1, 0 < x < (1-z)^2, sqrt(x) < y < 1-z

ccollins@math.utk.edu
Last Modified: April 20, 1999