# 15.6E: Exercises for Section 15.6

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In exercises 1 - 4, determine whether the statements are true or false.

1. If surface $$S$$ is given by $$\{(x,y,z) : \, 0 \leq x \leq 1, \, 0 \leq y \leq 1, \, z = 10 \}$$, then $$\displaystyle \iint_S f(x,y,z) \, dS = \int_0^1 \int_0^1 f (x,y,10) \, dx \, dy.$$

True

2. If surface $$S$$ is given by $$\{(x,y,z) : \, 0 \leq x \leq 1, \, 0 \leq y \leq 1, \, z = x \}$$, then $$\displaystyle \iint_S f(x,y,z) \, dS = \int_0^1 \int_0^1 f (x,y,x) \, dx \, dy.$$

3. Surface $$\vecs r = \langle v \, \cos u, \, v \, \sin u, \, v^2 \rangle,$$ for $$0 \leq u \leq \pi, \, 0 \leq v \leq 2$$ is the same surface $$\vecs r = \langle \sqrt{v} \, \cos 2u, \, \sqrt{v} \, \sin 2u, \, v \rangle,$$ for $$0 \leq u \leq \dfrac{\pi}{2}, \, 0 \leq v \leq 4$$.

True

4. Given the standard parameterization of a sphere, normal vectors $$t_u \times t_v$$ are outward normal vectors.

In exercises 5 - 10, find parametric descriptions for the following surfaces.

5. Plane $$3x - 2y + z = 2$$

$$\vecs r(u,v) = \langle u, \, v, \, 2 - 3u + 2v \rangle$$ for $$-\infty \leq u < \infty$$ and $$- \infty \leq v < \infty$$.

6. Paraboloid $$z = x^2 + y^2$$, for $$0 \leq z \leq 9$$.

7. Plane $$2x - 4y + 3z = 16$$

$$\vecs r(u,v) = \langle u, \, v, \, \dfrac{1}{3} (16 - 2u + 4v) \rangle$$ for $$|u| < \infty$$ and $$|v| < \infty$$.

8. The frustum of cone $$z^2 = x^2 + y^2$$, for $$2 \leq z \leq 8$$

9. The portion of cylinder $$x^2 + y^2 = 9$$ in the first octant, for $$0 \leq z \leq 3$$ $$\vecs r(u,v) = \langle 3 \, \cos u, \, 3 \, \sin u, \, v \rangle$$ for $$0 \leq u \leq \dfrac{\pi}{2}, \, 0 \leq v \leq 3$$

10. A cone with base radius $$r$$ and height $$h,$$ where $$r$$ and $$h$$ are positive constants.

For exercises 11 - 12, use a computer algebra system to approximate the area of the following surfaces using a parametric description of the surface.

11. [T] Half cylinder $$\{ (r, \theta, z) : \, r = 4, \, 0 \leq \theta \leq \pi, \, 0 \leq z \leq 7 \}$$

$$A = 87.9646$$

12. [T] Plane $$z = 10 - z - y$$ above square $$|x| \leq 2, \, |y| \leq 2$$

In exercises 13 - 15, let $$S$$ be the hemisphere $$x^2 + y^2 + z^2 = 4$$, with $$z \geq 0$$, and evaluate each surface integral, in the counterclockwise direction.

13. $$\displaystyle \iint_S z\, dS$$

$$\displaystyle \iint_S z \, dS = 8\pi$$

14. $$\displaystyle \iint_S (x - 2y) \, dS$$

15. $$\displaystyle \iint_S (x^2 + y^2) \, dS$$

$$\displaystyle \iint_S (x^2 + y^2) \, dS = 16 \pi$$

In exercises 16 - 18, evaluate $$\displaystyle \int \int_S \vecs F \cdot \vecs N \, dS$$ for vector field $$\vecs F$$ where $$\vecs N$$ is an outward normal vector to surface $$S.$$

16. $$\vecs F(x,y,z) = x\,\mathbf{\hat i}+ 2y\,\mathbf{\hat j} = 3z\,\mathbf{\hat k}$$, and $$S$$ is that part of plane $$15x - 12y + 3z = 6$$ that lies above unit square $$0 \leq x \leq 1, \, 0 \leq y \leq 1$$.

17. $$\vecs F(x,y) = x\,\mathbf{\hat i} + y\,\mathbf{\hat j}$$, and $$S$$ is hemisphere $$z = \sqrt{1 - x^2 - y^2}$$.

$$\displaystyle \iint_S \vecs F \cdot \vecs N \, dS = \dfrac{4\pi}{3}$$

18. $$\vecs F(x,y,z) = x^2\,\mathbf{\hat i} + y^2\,\mathbf{\hat j} + z^2\,\mathbf{\hat k}$$, and $$S$$ is the portion of plane $$z = y + 1$$ that lies inside cylinder $$x^2 + y^2 = 1$$. In exercises 19 - 20, approximate the mass of the homogeneous lamina that has the shape of given surface $$S.$$ Round to four decimal places.

19. [T] $$S$$ is surface $$z = 4 - x - 2y$$, with $$z \geq 0, \, x \geq 0, \, y \geq 0; \, \xi = x.$$

$$m \approx 13.0639$$

20. [T] $$S$$ is surface $$z = x^2 + y^2$$, with $$z \leq 1; \, \xi = z$$.

21. [T] $$S$$ is surface $$x^2 + y^2 + x^2 = 5$$, with $$z \geq 1; \, \xi = \theta^2$$.

$$m \approx 228.5313$$

22. Evaluate $$\displaystyle \iint_S (y^2 z\,\mathbf{\hat i}+ y^3\,\mathbf{\hat j} + xz\,\mathbf{\hat k}) \cdot dS,$$ where $$S$$ is the surface of cube $$-1 \leq x \leq 1, \, -1 \leq y \leq 1$$, and $$0 \leq z \leq 2$$ in a counterclockwise direction.

23. Evaluate surface integral $$\displaystyle \iint_S g \, dS,$$ where $$g(x,y,z) = xz + 2x^2 - 3xy$$ and $$S$$ is the portion of plane $$2x - 3y + z = 6$$ that lies over unit square $$R: 0 \leq x \leq 1, \, 0 \leq y \leq 1$$.

$$\displaystyle \iint_S g\,dS = 3 \sqrt{4}$$

24. Evaluate $$\displaystyle \iint_S (x + y + z)\, dS,$$ where $$S$$ is the surface defined parametrically by $$\vecs R(u,v) = (2u + v)\,\mathbf{\hat i} + (u - 2v)\,\mathbf{\hat j} + (u + 3v)\,\mathbf{\hat k}$$ for $$0 \leq u \leq 1$$, and $$0 \leq v \leq 2$$. 25. [T] Evaluate $$\displaystyle \iint_S (x - y^2 + z)\, dS,$$ where $$S$$ is the surface defined parametrically by $$\vecs R(u,v) = u^2\,\mathbf{\hat i} + v\,\mathbf{\hat j} + u\,\mathbf{\hat k}$$ for $$0 \leq u \leq 1, \, 0 \leq v \leq 1$$. $$\displaystyle \iint_S (x^2 + y - z) \, dS \approx 0.9617$$

26. [T] Evaluate where $$S$$ is the surface defined by $$\vecs R(u,v) = u\,\mathbf{\hat i} - u^2\,\mathbf{\hat j} + v\,\mathbf{\hat k}, \, 0 \leq u \leq 2, \, 0 \leq v \leq 1$$ for $$0 \leq u \leq 1, \, 0 \leq v \leq 2$$.

27. Evaluate $$\displaystyle \iint_S (x^2 + y^2) \, dS,$$ where $$S$$ is the surface bounded above hemisphere $$z = \sqrt{1 - x^2 - y^2}$$, and below by plane $$z = 0$$.

$$\displaystyle \iint_S (x^2 + y^2) \, dS = \dfrac{4\pi}{3}$$

28. Evaluate $$\displaystyle \iint_S (x^2 + y^2 + z^2) \, dS,$$ where $$S$$ is the portion of plane that lies inside cylinder $$x^2 + y^2 = 1$$.

29. [T] Evaluate $$\displaystyle \iint_S x^2 z \, dS,$$ where $$S$$ is the portion of cone $$z^2 = x^2 + y^2$$ that lies between planes $$z = 1$$ and $$z = 4$$. $$\text{div}\,\vecs F = a + b$$

$$\displaystyle \iint_S x^2 zdS = \dfrac{1023\sqrt{2\pi}}{5}$$

30. [T] Evaluate $$\displaystyle \iint_S \frac{xz}{y} \, dS,$$ where $$S$$ is the portion of cylinder $$x = y^2$$ that lies in the first octant between planes $$z = 0, \, z = 5$$, and $$y = 4$$. 31. [T] Evaluate $$\displaystyle \iint_S (z + y) \, dS,$$ where $$S$$ is the part of the graph of $$z = \sqrt{1 - x^2}$$ in the first octant between the $$xy$$-plane and plane $$y = 3$$. $$\displaystyle \iint_S (z + y) \, dS \approx 10.1$$

32. Evaluate $$\displaystyle \iint_S xyz\, dS$$ if $$S$$ is the part of plane $$z = x + y$$ that lies over the triangular region in the $$xy$$-plane with vertices (0, 0, 0), (1, 0, 0), and (0, 2, 0).

33. Find the mass of a lamina of density $$\xi (x,y,z) = z$$ in the shape of hemisphere $$z = (a^2 - x^2 - y^2)^{1/2}$$.

$$m = \pi a^3$$

34. Compute $$\displaystyle \int \int_S \vecs F \cdot \vecs N \, dS,$$ where $$\vecs F(x,y,z) = x\,\mathbf{\hat i} - 5y\,\mathbf{\hat j} + 4z\,\mathbf{\hat k}$$ and $$\vecs N$$ is an outward normal vector $$S,$$ where $$S$$ is the union of two squares $$S_1$$ : $$x = 0, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1$$ and $$S_2 \, : \, x = 0, \, 0 \leq x \leq 1, \, 0 \leq y \leq 1$$. 35. Compute $$\displaystyle \int \int_S \vecs F \cdot \vecs N \, dS,$$ where $$\vecs F(x,y,z) = xy\,\mathbf{\hat i} + z\,\mathbf{\hat j} + (x + y)\,\mathbf{\hat k}$$ and $$\vecs N$$ is an outward normal vector $$S,$$ where $$S$$ is the triangular region cut off from plane $$x + y + z = 1$$ by the positive coordinate axes.

$$\displaystyle \iint_S \vecs F \cdot \vecs N \, dS = \dfrac{13}{24}$$

36. Compute $$\displaystyle \int \int_S \vecs F \cdot \vecs N \, dS,$$ where $$\vecs F(x,y,z) = 2yz\,\mathbf{\hat i} + (\tan^{-1}xz)\,\mathbf{\hat j} + e^{xy}\,\mathbf{\hat k}$$ and $$\vecs N$$ is an outward normal vector $$S,$$ where $$S$$ is the surface of sphere $$x^2 + y^2 + z^2 = 1$$.

37. Compute $$\displaystyle \int \int_S \vecs F \cdot \vecs N \, dS,$$ where $$\vecs F(x,y,z) = xyz\,\mathbf{\hat i} + xyz\,\mathbf{\hat j} + xyz\,\mathbf{\hat k}$$ and $$\vecs N$$ is an outward normal vector $$S,$$ where $$S$$ is the surface of the five faces of the unit cube $$0 \leq x \leq 1, \, 0 \leq y \leq 1, \, 0 \leq z \leq 1$$ missing $$z = 0$$.

$$\displaystyle \iint_S \vecs F \cdot \vecs N \, dS = \dfrac{3}{4}$$

For exercises 38 - 39, express the surface integral as an iterated double integral by using a projection on $$S$$ on the $$yz$$-plane.

38. $$\displaystyle \iint_S xy^2 z^3 \, dS;$$ $$S$$ is the first-octant portion of plane $$2x + 3y + 4z = 12$$.

39. $$\displaystyle \iint_S (x^2 - 2y + z) \, dS;$$ $$S$$ is the portion of the graph of $$4x + y = 8$$ bounded by the coordinate planes and plane $$z = 6$$.

$$\displaystyle \int_0^8 \int_0^6 \left( 4 - 3y + \dfrac{1}{16} y^2 + z \right) \left(\dfrac{1}{4} \sqrt{17} \right) \, dz \, dy$$

For exercises 40 - 41, express the surface integral as an iterated double integral by using a projection on $$S$$ on the $$xz$$-plane.

40. $$\displaystyle \iint_S xy^2z^3 \, dS;$$ $$S$$ is the first-octant portion of plane $$2x + 3y + 4z = 12$$.

41. $$\displaystyle \iint_S (x^2 - 2y + z) \, dS;$$ is the portion of the graph of $$4x + y = 8$$ bounded by the coordinate planes and plane $$z = 6$$.

$$\displaystyle \int_0^2 \int_0^6 \big[x^2 - 2 (8 - 4x) + z\big] \sqrt{17} \, dz \, dx$$

42. Evaluate surface integral $$\displaystyle \iint_S yz \, dS,$$ where $$S$$ is the first-octant part of plane $$x + y + z = \lambda$$, where $$\lambda$$ is a positive constant.

43. Evaluate surface integral $$\displaystyle \iint_S (x^2 z + y^2 z) \, dS,$$ where $$S$$ is hemisphere $$x^2 + y^2 + z^2 = a^2, \, z \geq 0.$$

$$\displaystyle \iint_S (x^2 z + y^2 z) \, dS = \dfrac{\pi a^5}{2}$$

44. Evaluate surface integral $$\displaystyle \iint_S z \, dA,$$ where $$S$$ is surface $$z = \sqrt{x^2 + y^2}, \, 0 \leq z \leq 2$$.

45. Evaluate surface integral $$\displaystyle \iint_S x^2 yz \, dS,$$ where $$S$$ is the part of plane $$z = 1 + 2x + 3y$$ that lies above rectangle $$0 \leq x \leq 3$$ and $$0 \leq y \leq 2$$.

$$\displaystyle \iint_S x^2 yz \, dS = 171 \sqrt{14}$$

46. Evaluate surface integral $$\displaystyle \iint_S yz \, dS,$$ where $$S$$ is plane $$x + y + z = 1$$ that lies in the first octant.

47. Evaluate surface integral $$\displaystyle \iint_S yz \, dS,$$ where $$S$$ is the part of plane $$z = y + 3$$ that lies inside cylinder $$x^2 + y^2 = 1$$.

$$\displaystyle \iint_S yz \, dS = \dfrac{\sqrt{2}\pi}{4}$$

For exercises 48 - 50, use geometric reasoning to evaluate the given surface integrals.

48. $$\displaystyle \iint_S \sqrt{x^2 + y^2 + z^2} \, dS,$$ where $$S$$ is surface $$x^2 + y^2 + z^2 = 4, \, z \geq 0$$

49. $$\displaystyle \iint_S (x\,\mathbf{\hat i} + y\,\mathbf{\hat j}) \cdot dS,$$ where $$S$$ is surface $$x^2 + y^2 = 4, \, 1 \leq z \leq 3$$, oriented with unit normal vectors pointing outward

$$\displaystyle \iint_S (x\,\mathbf{\hat i} + y\,\mathbf{\hat j}) \cdot dS = 16 \pi$$

50. $$\displaystyle \iint_S (z\,\mathbf{\hat k}) \cdot dS,$$ where $$S$$ is disc $$x^2 + y^2 \leq 9$$ on plane $$z = 4$$ oriented with unit normal vectors pointing upward

51. A lamina has the shape of a portion of sphere $$x^2 + y^2 + z^2 = a^2$$ that lies within cone $$z = \sqrt{x^2 + y^2}$$. Let $$S$$ be the spherical shell centered at the origin with radius a, and let $$C$$ be the right circular cone with a vertex at the origin and an axis of symmetry that coincides with the $$z$$-axis. Determine the mass of the lamina if $$\delta(x,y,z) = x^2 y^2 z$$. $$m = \dfrac{\pi a^7}{192}$$

52. A lamina has the shape of a portion of sphere $$x^2 + y^2 + z^2 = a^2$$ that lies within cone $$z = \sqrt{x^2 + y^2}$$. Let $$S$$ be the spherical shell centered at the origin with radius a, and let $$C$$ be the right circular cone with a vertex at the origin and an axis of symmetry that coincides with the z-axis. Suppose the vertex angle of the cone is $$\phi_0$$, with $$0 \leq \phi_0 < \dfrac{\pi}{2}$$. Determine the mass of that portion of the shape enclosed in the intersection of $$S$$ and $$C.$$ Assume $$\delta(x,y,z) = x^2y^2z.$$ 53. A paper cup has the shape of an inverted right circular cone of height 6 in. and radius of top 3 in. If the cup is full of water weighing $$62.5 \, lb/ft^3$$, find the magnitude of the total force exerted by the water on the inside surface of the cup.

$$F \approx 4.57 \, lb$$

For exercises 54 - 55, the heat flow vector field for conducting objects i $$\vecs F = - k\vecs\nabla T$$, where $$T(x,y,z)$$ is the temperature in the object and $$k > 0$$ is a constant that depends on the material. Find the outward flux of $$\vecs F$$ across the following surfaces $$S$$ for the given temperature distributions and assume $$k = 1$$.

54. $$T(x,y,z) = 100 e^{-x-y}$$; $$S$$ consists of the faces of cube $$|x| \leq 1, \, |y| \leq 1, \, |z| \leq 1$$.

55. $$T(x,y,z) = - \ln (x^2 + y^2 + z^2)$$; $$S$$ is sphere $$x^2 + y^2 + z^2 = a^2$$.

$$8\pi a$$

For exercises 56 - 57, consider the radial fields $$\vecs F = \dfrac{\langle x,y,z \rangle}{(x^2+y^2+z^2)^{\dfrac{p}{2}}} = \dfrac{r}{|r|^p}$$, where $$p$$ is a real number. Let $$S$$ consist of spheres $$A$$ and $$B$$ centered at the origin with radii $$0 < a < b$$. The total outward flux across $$S$$ consists of the outward flux across the outer sphere $$B$$ less the flux into $$S$$ across inner sphere $$A.$$ 56. Find the total flux across $$S$$ with $$p = 0$$.

57. Show that for $$p = 3$$ the flux across $$S$$ is independent of $$a$$ and $$b.$$