MAT 267 (REVIEW)

TEST 3

Instr. S.Nikitin

Name_______________

ASU-ID#_______________

1. Evaluate the integral

$$\int_0^1 \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \int_{-\sqrt{1-x^2-y^2}}^{\sqrt{1-x^2-y^2}} \sqrt{x^2+y^2+z^2}dz dy dx$$

by changing coordinates.

2. Set up but do not evaluate, an iterated integration for

$$\int \int \int_B x^2+y^2 dV,$$

where $B$ lies above the plane $z=0,$ below the surface $z=4-x^2-y^2,$ and inside the cylinder $x^2+y^2=1.$

3. Set up but do not evaluate, an iterated integration for

$$\int \int \int_B z\sqrt{x^2+y^2+z^2}dV,$$

where $B$ is bounded by the hemisphere $z=\sqrt{4 - x^2 - y^2}$ and the $xy$-plane.

4. Is

$$F(x,y,z) = < 2\cdot xy, \;\;3 \cdot xy^2z,\;\; zy^2 >$$

a conservative vector field? Explain why.

5. Establish whether the vector field

$$F(x,y,z) = < \frac{1}{x},\;\;2e^{2y},y \cos(x) >$$

is conservative. If it is, then calculate its potential function.

6. Calculate the line integral

$$\int_C \nabla f \cdot dr,$$

where $f(x,y)=x^2 + x\cdot y^3,$ and $C$ is a curve between the points $(1,\;1)$ and $(-3,\;4).$

7. Find the work done by the force field

$$F(x,y) = < y^2,\;\; x-y >$$

on a particle that moves along the parabola $y=x^2$ between the points $(-2,\;4)$ and $(1,\;1).$

8. Use Green’s theorem to evaluate

$$\oint_C y^3dx - x^3dy,$$

where $C$ is the counter-clockwise oriented circle centered at the origin with radius $3.$

9. Calculate

$$\int_C yds,$$

where $C$ is the semicircle $x^2+y^2=9,\;\;y\ge 0.$

10. Let $F(x,y,z) = < xz,\;3yz,\;x^2y >.$ Find $div F (1,2,-3)$ and $curl F(x,y,z).$

11. Calculate the flux of $F(x,y,z) = <-y,\;x,\;z>$ across the surface $z=4-x^2-y^2$ (upward oriented), above the disk $x^2+y^2 \le 4.$