다중적분 전체 문제

Show that $\iiint_{R}f\left(\sqrt{16{x}^{2}+4{y}^{2}+{z}^{2}}\right)dV=\frac{\pi }{2}\int_{0}^{1}f(\rho ){\rho }^{2}d\rho ,$ where $f$ is a continuous function on $[0,1]$ and $R$ is the region bounded by the ellipsoid $16{x}^{2}+4{y}^{2}+{z}^{2}=1.$

[T] Find the area of the region bounded by the curves $xy=1,xy=3,y=2x,$ and $y=3x$ by using the transformation $u=xy$ and $v=\frac{y}{x}.$ Use a computer algebra system (CAS) to graph the boundary curves of the region $R.$

[T] Find the area of the region bounded by the curves $xy=2,xy=3,y=x,$ and $y=2x$ by using the transformation $u=xy$ and $v=\frac{y}{x}.$

Evaluate the triple integral $\int_{0}^{1}\,\int_{1}^{2}\,\int_{z}^{z+1}(y+1)dx\,dy\,dz$ by using the transformation $u=x-z,$ $v=3y,\text{and}\,w=\frac{z}{2}.$

Evaluate the triple integral $\int_{0}^{2}\,\int_{4}^{6}\,\int_{3z}^{3z+2}(5-4y)dx\,dz\,dy$ by using the transformation $u=x-3z,v=4y,\text{and}\,w=z.$

A transformation $T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2},T(u,v)=(x,y)$ of the form $x=au+bv,y=cu+dv,$ where $a,b,c,\text{and}\,d$ are real numbers, is called linear. Show that a linear transformation for which $ad-bc\ne 0$ maps parallelograms to parallelograms.

The transformation ${T}_{\theta }:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2},{T}_{\theta }(u,v)=(x,y),$ where $x=u\,\cos \,\theta -v\,\sin \,\theta ,$ $y=u\,\sin \,\theta +v\,\cos \,\theta ,$ is called a rotation of angle $\theta .$ Show that the inverse transformation of ${T}_{\theta }$ satisfies ${T}_{\theta }{}^{-1}={T}_{-\theta },$ where ${T}_{-\theta }$ is the rotation of angle $-\theta .$

[T] Use the results of the previous exercise to find the region $S$ in the $uv\text{-plane}$ whose image through a rotation of angle $\frac{\pi }{4}$ is the region $R$ enclosed by the ellipse ${x}^{2}+4{y}^{2}=1.$ Use a CAS to answer the following questions.

a. Graph the region $S.$ b. Evaluate the integral $\iint_{S}{e}^{-2uv}du\,dv.$ Round your answer to two decimal places.

[T] The transformations ${T}_{i}:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2},$ $i=1,\ldots ,\,4,$ defined by ${T}_{1}(u,v)=(u,-v),$ ${T}_{2}(u,v)=(-u,v),{T}_{3}(u,v)=(-u,-v),$ and ${T}_{4}(u,v)=(v,u)$ are called reflections about the $x\text{-axis},y\text{-axis},$ origin, and the line $y=x,$ respectively.

a. Find the image of the region $S=\{(u,v)|{u}^{2}+{v}^{2}-2u-4v+1\le 0\}$ in the $xy\text{-plane}$ through the transformation ${T}_{1}\circ {T}_{2}\circ {T}_{3}\circ {T}_{4}.$ b. Use a CAS to graph the image of $S.$ c. Evaluate the integral $\iint_{S}\sin ({u}^{2})du\,dv$ by using a CAS. Round your answer to two decimal places.

[T] The transformation ${T}_{k,1,1}:{\mathbb{R}}^{3}\to {\mathbb{R}}^{3},{T}_{k,1,1}(u,v,w)=(x,y,z)$ of the form $x=ku,$ $y=v,z=w,$ where $k\ne 1$ is a positive real number, is called a stretch if $k>1$ and a compression if $0<k<1$ in the $x\text{-direction}.$ Use a CAS to evaluate the integral $\iiint_{S}{e}^{-(4{x}^{2}+9{y}^{2}+25{z}^{2})}dx\,dy\,dz$ on the solid $S=\{(x,y,z)|4{x}^{2}+9{y}^{2}+25{z}^{2}\le 1\}$ by considering the compression ${T}_{2,3,5}(u,v,w)=(x,y,z)$ defined by $x=\frac{u}{2},y=\frac{v}{3},$ and $z=\frac{w}{5}.$ Round your answer to four decimal places.

[T] The transformation ${T}_{a,0}:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2},{T}_{a,0}(u,v)=(u+av,v),$ where $a\ne 0$ is a real number, is called a shear in the $x\text{-direction}.$ The transformation, ${T}_{0,b}:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2},{T}_{0,b}(u,v)=(u,bu+v),$ where $b\ne 0$ is a real number, is called a shear in the $y\text{-direction}.$

a. Find transformations ${T}_{0,2}\circ {T}_{3,0}.$ b. Find the image $R$ of the trapezoidal region $S$ bounded by $u=0,v=0,v=1,$ and $v=2-u$ through the transformation ${T}_{0,2}\circ {T}_{3,0}.$ c. Use a CAS to graph the image $R$ in the $xy\text{-plane}.$ d. Find the area of the region $R$ by using the area of region $S.$

Use the transformation, $x=au,y=av,z=cw$ and spherical coordinates to show that the volume of a region bounded by the spheroid $\frac{{x}^{2}+{y}^{2}}{{a}^{2}}+\frac{{z}^{2}}{{c}^{2}}=1$ is $\frac{4\pi {a}^{2}c}{3}.$

Find the volume of a football whose shape is a spheroid $\frac{{x}^{2}+{y}^{2}}{{a}^{2}}+\frac{{z}^{2}}{{c}^{2}}=1$ whose length from tip to tip is $11$ inches and circumference at the center is $22$ inches. Round your answer to two decimal places.

[T] Lamé ovals (or superellipses) are plane curves of equations ${\left(\frac{x}{a}\right)}^{n}+{\left(\frac{y}{b}\right)}^{n}=1,$ where a, b, and n are positive real numbers.

a. Use a CAS to graph the regions $R$ bounded by Lamé ovals for $a=1,b=2,n=4$ and $n=6,$ respectively. b. Find the transformations that map the region $R$ bounded by the Lamé oval ${x}^{4}+{y}^{4}=1,$ also called a squircle and graphed in the following figure, into the unit disk. c. Use a CAS to find an approximation of the area $A(R)$ of the region $R$ bounded by ${x}^{4}+{y}^{4}=1.$ Round your answer to two decimal places.

[T] Lamé ovals have been consistently used by designers and architects. For instance, Gerald Robinson, a Canadian architect, has designed a parking garage in a shopping center in Peterborough, Ontario, in the shape of a superellipse of the equation ${\left(\frac{x}{a}\right)}^{n}+{\left(\frac{y}{b}\right)}^{n}=1$ with $\frac{a}{b}=\frac{9}{7}$ and $n=e.$ Use a CAS to find an approximation of the area of the parking garage in the case $a=900$ yards, $b=700$ yards, and $n=2.72$.

Chapter Review Exercises

True or False? Justify your answer with a proof or a counterexample.

$\int_{a}^{b}\,\int_{c}^{d}f(x,y)dy\,dx=\int_{c}^{d}\,\int_{a}^{b}f(x,y)dy\,dx$

Fubini’s theorem can be extended to three dimensions, as long as $f$ is continuous in all variables.

The integral $\int_{0}^{2\pi }\,\int_{0}^{1}\,\int_{r}^{1}dz\,dr\,d\theta$ represents the volume of a right cone.

The Jacobian of the transformation $x={u}^{2}-2v,y=3v-2uv$ is given by $-4{u}^{2}+6u+4v.$

Evaluate the following integrals.

$\iint_{R}(5{x}^{3}{y}^{2}-{y}^{2})dA,R=\{(x,y)|0\le x\le 2,1\le y\le 4\}$