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Exercise 1: Integration by Parts and Substitution with Two Variables
A
Compute
$$\displaystyle{\int_0^{\frac{\pi}{2}}\left(\int_0^{\frac{\pi}{2}} u\cos(u+v)\,\mathrm du\right)\,\mathrm dv\,.}$$
Show hint
Find by integration by parts with respect to $\,u\,$ an antiderivative $\,F(u)\,$ to the function
$$\,u\cos(u+v)\,.$$
Then calculate $\displaystyle{F\left(\frac{\pi}{2}\right)-F(0)}$ which is a function of $\,v\,$ for which you now must in turn find an antiderivative $\,G(v)\,$ .
Show hint
$$G(v)=\frac{\pi}{2}\,\cos(v)-\sin(v)-\cos(v)$$
Show answer
$$\displaystyle{\frac{\pi}{2}-2}$$
B
Compute
$$\displaystyle{\int_0^1\left(\int_0^1\, \frac{v}{(uv+1)^2}\,\mathrm du\right)\,\mathrm dv\,.}$$
Show hint
An antiderivative to the contents of the outer integral becomes
$$\displaystyle{G(v)=1-\frac{1}{v+1}}\,.$$
Exercise 2: Plane Integral with Parametrization We will compute the plane integral
$$\int_B \,2xy\,\mathrm d\mu\, \quad\mathrm{where}\quad B=\left\lbrace (x,y)\,\vert\, 0\leq x\,, 0\leq y\,, x+y\leq 1\right\rbrace\,.$$
Follow the steps below.
C
Firstly, sketch region $B.$
D
Secondly, create a parametric representation of $B\,$ .
Show hint
See the parametrization examples in today’s Maple demo.
Show answer
One possibility is
$$\mathbf r(u,v)=(u,v(1-u))\quad \mathrm{where}\quad u\in\left[ 0,1\right]\,\,\mathrm{and}\,\, v\in\left[ 0,1\right]\,.$$
E
Thirdly, compute the Jacobian function that corresponds to your parametrization.
Show answer
$$\text{Jac}_\mathbf r(u,v)=1-u$$
F
Fourthly, determine the restriction of the functino $f$ to the parametrization, $f(\mathbf r(u,v))$ .
G
Fifth and finally, compute the given plane integral.
Show hint
Multiply the restriction of $f$ to the parametrization with the Jacobian.
Show hint
$$\int_0^1\left(\int_0^1 2u^3v-4u^2v+2uv\;\mathrm du\right)\,\mathrm dv$$
Show hint
An antiderivative with respect to $\,u\,$ is:
$$\displaystyle{\frac 12\,u^4v-\frac 43\,u^3v+u^2v}\,.$$
Show hint
After substituting in the limits for $\,u\,$ , you will get the new integrand for the outer integral
$$\,\displaystyle{\frac 16\, v}\,.$$
Show answer
$$\int_B \,2xy \,\mathrm d\mu=\frac{1}{12}$$
Exercise 3: Graph Surface, Parametrization and Integration If a function of two variables defined over an axis-parallel rectangle is given as:
$$h(x,y)\,,\,\, x\in\left[ a,b\right] ,\,y\in\left[ c,d\right]\,,$$
then it is fairly easy to state a parametric representation of its graph:
$$\begin{matr}{c}x\newline y\newline z\end{matr}=\mr(u,v)=\begin{matr}{c}u\newline v\newline h(u,v)\end{matr}
\,,\,\, u\in\left[ a,b\right] ,\,v\in\left[ c,d\right]\,.$$
Use this observation in the following question.
A
We are given the function $\,h(x,y)=\sqrt 3\,y\,$ and the set of points
$$\,M=\left\{(x,y)\,|\,x\in\left[ 0,1\right] ,\,y\in\left[ 0,2\right]\right\}\,.$$
Let $G$ denote the part of the graph of $h$ that is located vertically above $M\,.$
State a parametric representation of this elevated surface $G\,,$ and compute the surface integral
$$\int_G\,xyz\,\mathrm d\mu\,.$$
Show hint
A parametric representation is
$$\mr(u,v)=(u,v,\sqrt 3\,v)\,,\,\, u\in\left[ 0,1\right] ,\,v\in\left[ 0,2\right]\,.$$
Show hint
$$\mathrm{Jac}_\mathbf{r}(u,v)=\,|\,\mathbf N(u,v)\,|=|\,\mathbf r'_u(u,v)\times \mathbf r'_v(u,v)\,|\,$$
Show hint
$$\displaystyle{\mathbf N(u,v)=
\begin{matr}{c}1\newline 0\newline 0\end{matr}\times
\begin{matr}{c}0\newline \sqrt 3\newline 1\end{matr}=
\begin{matr}{c}0\newline -\sqrt 3\newline 1\end{matr}}$$
$\mathrm{Jac}_\mathbf{r}(u,v)$ is the length of this vector.
Show hint
$$\mathrm{Jac}_\mathbf{r}(u,v)=2$$
Show hint
To set up the integrand for integration, multiply $f(\mathbf r(u,v))$ with $\mathrm{Jac}_\mathbf{r}(u,v)$ . Compute $f(\mathbf r(u,v))$ by substituting in the three coordinates of $\mathbf r$ for $x$ , $y$ and $z$ in $f$ .
Show answer
$$\int_Gxyz\,\mathrm d\mu=\frac{8\sqrt 3}{3}$$
Exercise 4: Surface of Revolution A parabola segment $K$ in the $(x,z)$ plane is given by the equation
$$z=\frac{x^2}4\,\,,\,\,x\in [\,0\,,\,2\,]\,.$$
A
Explain why $K$ when considered as a space curve can be described by the parametric representation
$$\mr(u)=(g(u),0,h(u))=\left(u,0,\frac{u^2}{4}\right), \quad\mathrm{where}\,u\in\left[\, 0,2\,\right] .$$
Show hint
The second coordinate $0$ only accounts for the fact that $y=0\,$ since the curve is located in the $(x,z)$ plane.
We will now consider $K$ to be a profile curve. A surface of revolution $F$ appears from rotating $K$ an angle of $2\pi$ about the $z$ axis.
B
Explain that $F$ can be described by the parametric representation
$$\begin{align*}
\mr(u,v)&=\big(\,g(u)\cos(v),g(u)\sin(v),h(u)\,\big)\newline
&=\big(\,u\cos(v),u\sin(v),\frac{u^2}{4}\,\big)\,,\,\,u\in[\,0,2\,]\,\,,v\in[\,0,2\pi\,]\,.
\end{align*}$$
Show hint
See expression (23.30) in eNote 23, Section 23.1.2. Or see today’s Maple demo.
The parametric representation appears by multiplying the rotation matrix
$$\begin{matr}{ccc}
\cos(v)&-\sin(v)&0\newline
\sin(v)&\cos(v)&0\newline
0&0&1\end{matr}\,$$
with the parametric representation of $K\,$ .
C
We now introduce the function $f(x,y,z)=x^2+y^2\,.$ Compute the surface integral
$$\int_F f\,\mathrm d\mu.$$
Show answer
$$\frac{64(1+\sqrt 2)\pi }{15}$$
Exercise 5: Graph Surface For a function of two variables
$$h(x,y)=2-x^2-y^2\,$$
we consider the following two graph surfaces:
$$\begin{align*}
F&=\lbrace(x,y,z)\,\vert\,x\in \left[\,0\,,\,1\,\right]\,, y\in \left[\,0\,,\,2\,\right] \,\,\mathrm{and}\,\, z=h(x,y)\,\rbrace\,,\newline
G&=\lbrace(x,y,z)\,\vert\, x^2+y^2\leq 2\,\,\mathrm{and}\,\, z=h(x,y)\,\rbrace\,.
\end{align*}$$
A
Compute
$$\displaystyle{\int_F\sqrt{9-4z}\,\mathrm d\mu}\,.$$
Show hint
Note that $\,F\,$ is an axis-parallel rectangle in the $(x,y)$ plane. You can easily “simulate” a plot of $F$ by plotting $h$ , limiting its parameter ranges to those given, and then changing orientation so you view it “from above”. This will help you to create a parametric representation of $\,F\,.$
Show hint
The integrand becomes simpler than you would expect, because the parametric representation substituted into the function becomes equal to the Jacobian. The square root disappears!
Show hint
$$\int_0^2\left(\int_0^1 1+4u^2+4v^2\,du\right)\,\mathrm dv$$
Show answer
$$\int_F\sqrt{\,9-4z}\,\mathrm d\mu=\,\frac {46}3$$
B
Compute
$$\displaystyle{\int_G\sqrt{9-4z}\,\mathrm d\mu}\,.$$
Show hint
This time the region in the $(x,y)$ plane is not an axis-parallel rectangle. First create a parametric representation of the region $\,x^2+y^2\leq 2\,$ in the $\,(x,y)$ plane, and then set up a parametric representation of $\,G\,.$
Show hint
The region in the $(x,y)$ plane is a circular disc. It can be parametrized by
$$(x,y)=(u\cos (v),u\sin (v))\,,\,\, u\in [\,0,\sqrt 2\,]\,,\,\,v\in [\,0,2\pi\,]\,.$$
Show hint
$\,G\,$ can be parametrized by
$$\,(x,y,z)=\mr(u,v)=(u\cos (v),u\sin (v),2-u^2)\,,\,\, u\in [\,0,\sqrt 2\,]\,,\,\,v\in [\,0,2\pi\,]\,.$$
The third coordinate of $\,G\,$ appears after having used the “dummy rule”.
Show hint
Find the Jacobian, and compute the integral - possibly use Maple to find antiderivatives.
Show answer
$$\int_G\sqrt{9-4z}\,\mathrm d\mu=10\pi$$
