21-259  Calculus in 3-D  Spring 2003

Review Problems for Test 3

1. Evaluate each of the following.

(a) ∫_{0}^{1} ∫_{0}^{y} (x^{2} + y^{2}) dxdy

(b) ∫_{−2}^{2} ∫_{0}^{√\overline{4−x^{2}}}e^{x^{2}+y^{2}} dydx

2. Let D be the region in the first quadrant that is bounded by y = \sqrt{1 − x^{2}}, y = x, and the x-axis. Evaluate

∫D∫ (1 + \sqrt{x^{2} + y^{2}}) dA

3. Find the volume of the region enclosed by y = x^{2}, y = 1, z = 0, and z = 2.

4. Find the volume of the region that is inside the sphere x^{2} + y^{2} + z^{2} = 4, but outside the cylinder x^{2} + y^{2} = 1.

5. Let E be the region in the first octant that is bounded by the coordinate planes, the plane x + y = 1 and the plane z = 3. Evaluate

∫ _{∫E}∫ (1 + 2z)dV.

6. Let E be the region in the first octant that is bounded by the coordinate planes and the sphere x^{2} + y^{2} + z^{2} = 1. Evaluate

∫ _{∫E}∫ (x^{2} + y^{2} + z^{2})^{3/2}dV.

7. Evaluate each of the following line integrals.

(a) _{∫C} F⃗ · dr⃗, where F⃗ (x, y, z) = y i⃗ − x j⃗ + (x^{2} + y^{2} + z^{2})k⃗ and C is described

by r⃗(t) = sin i⃗ + cos t j⃗ + t k⃗, 0 ≤ t ≤ π.

(b) _{∫C} (∇f) · dr⃗, where f(x, y, z) = xyz and C is described by

r⃗(t) = (1 + t^{4})i⃗ + t^{3} j⃗ + e^{t} k⃗, 0 ≤ t ≤ 1.

(c) _{∫C} F⃗ · dr⃗, where F⃗ (x, y, z) = (yz + x)i⃗ + xz j⃗ + xy k⃗, and C is described

by r⃗(t)t = t i⃗ + t^{2} j⃗ + t k⃗, 0 ≤ t ≤ 1.

8. Evaluate each of the following line integrals.

(a) _{∫C}ydx + xydy, where C is the segment of y = x^{3} from (0, 0) to (1, 1).

(b) _{∫C} − ydx + (x + y^{2})dy, where C is the segment of x^{2} + y^{2} = 1 from (1,0)

to (0,1).

9. Find a function f(x, y) such that ∇f(x, y) = (2xy + x)i⃗ + (x^{2} + y^{3})j⃗.

10. Evaluate

∫C(e^{y} + xy)dx + xe^{y}dy,

where C is the segment of y = x^{2} from (0,0) to (1,1).

11. Let C be a piecewise smooth simple closed curve in the x − y plane and let D be the region enclosed by C. Given that

∫D∫ dA = π,_{∫D}∫ ydA = 3, _{∫C}x^{2}dy = 2,

evaluate

∫C(xy + 2y + 1)dx + (y^{2} + 3xy + 5x)dy.

12. Use Green’s Theorem to evaluate _{∫C}y^{3}dx − x^{3}dy where C is the circle x^{2} + y^{2} = 1. In Problems 11-13, C is oriented counterclockwise.

13. Let C be a piecewise smooth simple closed curve in the x−y plane and let D be the region enclosed by C. Show that if u(x, y) is a smooth function satisfying u_{xx} + u_{yy} = 0 in D then

∫D∫ (u_{x}^{2} + u_{y}^{2}) dA = _{∫C} − uu_{y}dx + uu_{x}dy.

14. Let S be the portion of the cone z = 4 − \sqrt{x^{2} − y^{2}} for which 1 ≤ x^{2} + y^{2} ≤ 4. (a) Find the surface area of S.

(b) Evaluate _{∫S}∫ (z + 2\sqrt{x^{2} + y^{2}}) dS 15. Let S be the hemisphere z = \sqrt{9 − x^{2} − y^{2}} and let n⃗ be the unit normal to S

that points upward. Evaluate

∫S∫ F⃗ · n⃗dS,

where F⃗ (x, y, z) = x i⃗ + y j⃗ + z k⃗.