일반 영역의 이중적분

[T] The Reuleaux triangle consists of an equilateral triangle and three regions, each of them bounded by a side of the triangle and an arc of a circle of radius s centered at the opposite vertex of the triangle. Show that the area of the Reuleaux triangle in the following figure of side length $s$ is $\frac{{s}^{2}}{2}\left(\pi -\sqrt{3}\right).$

[T] Show that the area of the lunes of Alhazen, the two blue lunes in the following figure, is the same as the area of the right triangle ABC. The outer boundaries of the lunes are semicircles of diameters $AB\,\text{and}\,BC,$ respectively, and the inner boundaries are formed by the circumcircle of the triangle $ABC.$

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.

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

Evaluate the following integrals.

$\iint_{D}\frac{y}{3{x}^{2}+1}dA,D=\{(x,y)|0\le x\le 1,-x\le y\le x\}$

$\iint_{D}\sin ({x}^{2}+{y}^{2})dA$ where $D$ is a disk of radius $2$ centered at the origin

Evaluate the following integrals.

$\int_{0}^{1}\,\int_{y}^{1}xy{e}^{{x}^{2}}dx\,dy$

Evaluate the following integrals.

$\int_{-1}^{1}\,\int_{0}^{z}\,\int_{0}^{x-z}6dy\,dx\,dz$

Evaluate the following integrals.

$\iiint_{R}3y\,dV,$ where $R=\{(x,y,z)|0\le x\le 1,0\le y\le x,0\le z\le \sqrt{9-{y}^{2}}\}$

Evaluate the following integrals.

$\int_{0}^{2}\,\int_{0}^{2\pi }\,\int_{r}^{1}r\,dz\,d\theta \,dr$

Evaluate the following integrals.

$\int_{0}^{2\pi }\,\int_{0}^{\pi /2}\,\int_{1}^{3}{\rho }^{2}\sin (\varphi )d\rho \,d\varphi \,d\theta$

Evaluate the following integrals.

$\int_{0}^{1}\,\int_{-\sqrt{1-{x}^{2}}}^{\sqrt{1-{x}^{2}}}\,\int_{-\sqrt{1-{x}^{2}-{y}^{2}}}^{\sqrt{1-{x}^{2}-{y}^{2}}}dz\,dy\,dx$

The area of region enclosed by one petal of $r=\cos (4\theta ).$

The volume of the solid that lies between the paraboloid $z=2{x}^{2}+2{y}^{2}$ and the plane $z=8.$

The volume of the solid bounded by the cylinder ${x}^{2}+{y}^{2}=16$ and from $z=1$ and $z+x=2.$

The volume of the intersection between two spheres of radius 1, the top whose center is $(0,0,0.25)$ and the bottom, which is centered at $(0,0,0).$

$\rho (x,y)=xy$ on the circle with radius $1$ in the first quadrant only.