Question 17.117

Evaluate the integral of the function \(f(x,y,z) = x+y\) over the surface \(S\) given by: \[ \Phi(u,v) = ( 2u \cos v, 2u \sin v, u ), \quad u \in [0,4], v \in [0, \pi] \]

Question 17.118

Evaluate the integral of the function \(f(x,y,z) = z+6\) over the surface \(S\) given by: \[ \Phi(u,v) = ( u, \frac{v}{3}, v ), \quad u \in [0,2], v \in [0, 3]. \]

Question 17.119

Evaluate the integral \[ {\int\!\!\!\int}_{S} (3x - 2y + z) \,dS, \] where \(S\) is the portion of the plane \(2x + 3y + z = 6\) that lies in the first octant.

Question 17.120

Evaluate the integral \[ {\int\!\!\!\int}_{S} (x+z) \,dS, \] where \(S\) is the part of the cylinder \(y^2 + z^2 =4\) with \(x \in [0,5]\).

Question 17.121

Let \(S\) be the surface defined by \[ \Phi(u,v) = ( u+v, u-v, uv ). \]

• (a) Show that the image of \(S\) is in the graph of the surface \(4z = x^2 - y^2\).
• (b) Evaluate \({\int\!\!\!\int}_S x \,dS\) for all points on the graph \(S\), over \(x^2 + y^2 \leq 1\).

Question 17.122

Evaluate the integral \[ {\int\!\!\!\int}_{S} (x^{2}z + y^{2}z) \,dS, \] where \(S\) is the part of the plane \(z=4+x+y\) that lies inside the cylinder \(x^{2}+y^{2}=4\).

Question 17.123

Compute \({\int\!\!\!\int}_S {\it xy} \,dS\), where \(S\) is the surface of the tetrahedron with sides \(z=0, y=0\), \(x + z = 1\), and \(x = y\).

Question 17.124

Evaluate \({\int\!\!\!\int}_S {\it xyz} \,dS\), where \(S\) is the triangle with vertices \((1,0, 0), (0, 2, 0)\), and \((0, 1, 1)\).

Question 17.125

Evaluate \({\int\!\!\!\int}_S z \,dS\), where \(S\) is the upper hemisphere of radius \(a\), that is, the set of \((x, y, z)\) with \(z = \sqrt{a^2 - x^2 - y^2}\).

Question 17.126

Evaluate \({\int\!\!\!\int}_S (x+y+z)\,dS\), where \(S\) is the boundary of the unit ball \(B \); that is, \(S\) is the set of \((x, y, z)\) with \(x^2 + y^2 + z^2 = 1\). (HINT: Use the symmetry of the problem.)

Question 17.127

• (a) Compute the area of the portion of the cone \(x^2 + y^2 = z^2\) with \(z \ge 0\) that is inside the sphere \(x^2 + y^2 + z^2 =2Rz\), where \(R\) is a positive constant.
• (b) What is the area of that portion of the sphere that is inside the cone?

Question 17.128

Verify that in spherical coordinates, on a sphere of radius R, \[ \|{\bf T}_\phi\times {\bf T}_\theta \|\, d\phi\, d\theta=R^2\sin\phi\ d\phi\ d\theta. \]

Question 17.129

Evaluate \({\int\!\!\!\int}_S z \,dS\), where \(S\) is the surface \(z = x^2 + y^2, x^2 + y^2 \le 1\).

Question 17.130

Evaluate the surface integral \({\int\!\!\!\int}_S z^2 \,dS\), where \(S\) is the boundary of the cube \(C=[-1,1]\times [-1,1] \times [-1,1]\). (HINT: Do each face separately and add the results.)

Question 17.131

Find the mass of a spherical surface \(S\) of radius \(R\) such that at each point \((x, y, z) \in S\) the mass density is equal to the distance of \((x, y, z)\) to some fixed point \((x_0, y_0, z_0)\in S\).

Question 17.132

A metallic surface \(S\) is in the shape of a hemisphere \(z \,{=}\, \sqrt{R^2 \,{-}\,x^2 \,{-}\, y^2}\), where \((x, y)\) satisfies \(0 \le x^2 \,{+}\,y^2 \le R^2\). The mass density at \((x, y, z)\in S\) is given by \(m(x,y,z) = x^2 + y^2\). Find the total mass of \(S\).

Question 17.133

Let \(S\) be the sphere of radius \(R\).

• (a) Argue by symmetry that \[ \int\!\!\!\int\nolimits_{S} x^2 \,dS = \int\!\!\!\int\nolimits_{S} y^2 \,dS = \int\!\!\!\int\nolimits_{S} z^2 \,dS. \]
• (b) Use this fact and some clever thinking to evaluate, with very little computation, the integral \[ \int\!\!\!\int\nolimits_{S} x^2 \,dS. \]
• (c) Does this help in Exercise 16?

Question 17.134

• (a) Use Riemann sums to justify the formula \[ \frac{1}{A(S\,)} \int\!\!\!\int\nolimits_{S} f(x,y,z)\,dS \] for the average value of \(f\) over the surface \(S\).
• (b) In Example 3 of this section, show that the average of \(f(x,y,z) = z^2\) over the sphere is \(1/3\).
• (c) Define the center of gravity \((\bar{x},\bar{y},\bar{z})\) of a surface \(S\) to be such that \(\bar{x},\bar{y}\), and \(\bar{z}\) are the average values of the \(x, y\), and \(z\) coordinates on \(S\). Show that the center of gravity of the triangle in Example 4 of this section is \((\frac13,\frac13,\frac13)\).

399

Question 17.135

Find the average value of \(f(x,y,z) = x + z^2\) on the truncated cone \(z^{2} = x^{2}+y^{2}\), with \(1 \leq z \leq 4\).

Question 17.136

Evaluate the integral \[ {\int\!\!\!\int}_{S} (1-z) \,dS, \] where \(S\) is the graph of \(z=1 - x^{2} - y^{2}\), with \(x^{2}+y^{2} \leq 1\).

Question 17.137

Find the \(x, y\), and \(z\) coordinates of the center of gravity of the octant of the solid sphere of radius \(R\) and centered at the origin determined by \(x \ge 0 , y \ge 0, z \ge 0\). (HINT: Write this octant as a parametrized surface—see Example 3 of this section and an exercise above.)

Question 17.138

Find the \(z\) coordinate of the center of gravity (the average \(z\) coordinate) of the surface of a hemisphere \((z \le 0)\) with radius \(r\) (see an exercise above). Argue by symmetry that the average \(x\) and \(y\) coordinates are both zero.

Question 17.139

Let \({\Phi}{:}\, D \subset {\mathbb R}^{2} \to {\mathbb R}^{3}\) be a parametrization of a surface \(S\) defined by \[ x=x(u, v),\qquad y=y(u, v),\qquad z=z(u, v). \]

• (a) Let \[ \frac{\partial{\Phi}}{\partial u}=\bigg(\frac{\partial x}{\partial u}, \frac{\partial y}{\partial u}, \frac{\partial z}{\partial u}\bigg) \quad \hbox{and}\quad \frac{\partial{\Phi}}{\partial v} = \bigg(\frac{\partial x}{\partial v}, \frac{\partial y}{\partial v}, \frac{\partial z}{\partial v}\bigg), \] that is, \(\partial {\Phi}/\partial u = {\bf T}_{u}\) and \(\partial {\Phi}/\partial v = {\bf T}_{v}\), and set \[ E=\Big\|\frac{\partial{\Phi}}{\partial u}\Big\|^2,\qquad F = \frac{\partial{\Phi}}{\partial u} \ {\cdot} \ \frac{\partial{\Phi}}{\partial v}, \qquad G =\Big\|\frac{\partial{\Phi}}{\partial v}\Big\|^2. \] Show that \[ \sqrt{EG - F^2}=\|{\bf T}_u\times {\bf T}_v\|, \] and that the surface area of \(S\) is \[ A(S)=\int\!\!\int_{D} \sqrt{EG - F^{2}} \, {du\, dv}. \] In this notation, how can we express \(\int\!\!\int_{S} {\it f dS}\) for a general function of \(f\)?
• (b) What does the formula for \(A(S)\) become if the vectors \(\partial {\Phi}/\partial u\) and \(\partial {\Phi}/\partial v\) are orthogonal?
• (c) Use parts (a) and (b) to compute the surface area of a sphere of radius \(a\).

Question 17.140

Dirichlet’s functional for a parametrized surface \({\Phi}\colon\, D\to {\mathbb R}^3\) is defined byfootnote # \[ J({\Phi}) = \frac{1}{2} \int\!\!\!\int\nolimits_{D} \Big(\Big\|\frac{\partial{\Phi}}{\partial u}\Big\|^2+ \Big\|\frac{\partial{\Phi}}{\partial v}\Big\|^2\Big)\, du \,dv. \]

Use Exercise 23 to argue that the area \(A({\Phi}) \le J({\Phi})\) and equality holds if \[ \hbox{(a)} \Big\|\frac{\partial {\Phi}}{\partial u}\Big\|^2 = \Big\|\frac{\partial{\Phi}}{\partial v}\Big\|^2\qquad \hbox{and }\qquad \hbox{(b)} \frac{\partial{\Phi}}{\partial u}\, {\cdot}\, \frac{\partial{\Phi}}{\partial v} = 0. \]

Compare these equations with the previous exercise and the remarks at the end of Section 17.4. A parametrization \({\Phi}\) that satisfies conditions (a) and (b) is said to be conformal.

Question 17.141

Let \(D\subset{\mathbb R}^2\) and \({\Phi}\colon\, D \to {\mathbb R}^2\) be a smooth function \({\Phi}(u,v) = (x(u,v),y(u,v))\) satisfying conditions (a) and (b) of Exercise 16 and assume that \begin{eqnarray*} \hbox{ det } \left[\begin{array}{l@{\qquad}l} \\[-8pt] \displaystyle\frac{\partial x}{\partial u} & \displaystyle\frac{\partial x}{\partial v}\\[11pt] \displaystyle\frac{\partial y}{\partial u} &\displaystyle\frac{\partial y}{\partial v}\\[7pt] \end{array}\right] >0.\\[-4pt] \end{eqnarray*}

Show that \(x\) and \(y\) satisfy the Cauchy–Riemann equations \(\partial x/\partial u = \partial y/\partial v, \partial x/\partial v = - \partial y/\partial u\).

Conclude that \(\nabla^2{\Phi} = 0\) (i.e., each component of \({\Phi}\) is harmonic).

Question 17.142

Let \(S\) be a sphere of radius \(r\) and \({\bf p}\) be a point inside or outside the sphere (but not on it). Show that \[ \int\!\!\!\int\nolimits_{S}\frac{1}{\|{\bf x}-{\bf p}\|} \,dS = \left\{ \begin{array}{l@{\qquad}l} 4\pi r &\hbox{if}\quad{\bf p}\hbox{ is inside } S\\ 4\pi r^2/d &\hbox{if}\quad{\bf p}\hbox{ is outside } S, \end{array}\right. \] where \(d\) is the distance from \({\bf p}\) to the center of the sphere and the integration is over the sphere. [HINT: Assume \({\bf p}\) is on the \(z\)-axis. Then change variables and evaluate. Why is this assumption on \({\bf p}\) justified?]

400

Question 17.143

Find the surface area of that part of the cylinder \(x^2 + z^2 = a^2\) that is inside the cylinder \(x^2 + y^2 = 2ay\) and also in the positive octant \((x \ge 0, y \ge 0, z \ge 0)\). Assume \(a > 0\).

Question 17.144

Let a surface \(S\) be defined implicitly by \(F(x, y, z) = 0\) for \((x, y)\) in a domain \(D\) of \({\mathbb R}^2\). Show that \begin{eqnarray*} &&\int\!\!\!\int\nolimits_{S} \Big|\frac{\partial F}{\partial z}\Big|\,dS \\[4pt] &&= \int\!\!\!\int\nolimits_{D} \sqrt{\Big(\frac{\partial F}{\partial x}\Big)^2 + \Big(\frac{\partial F}{\partial y}\Big)^2 + \Big(\frac{\partial F}{\partial z}\Big)^2} \,{\it dx} \,{\it dy}. \end{eqnarray*}

Compare with Exercise 22 of Section 17.4.