Question 1.117

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 1.118

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 1.119

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 1.120

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 1.121

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 1.122

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 1.123

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 1.124

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 1.125

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 1.126

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 1.127

• (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 1.128

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 1.129

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

Question 1.130

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 1.131

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 1.132

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 1.133

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 1.134

• (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 1.135

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 1.136

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 1.137

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 1.138

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 1.139

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 1.140

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 1.5. A parametrization \({\Phi}\) that satisfies conditions (a) and (b) is said to be conformal.

Question 1.141

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 1.142

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 1.143

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 1.144

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 1.5.