일반 영역의 이중적분

Evaluate the iterated integrals.

$\int_{0}^{1}\,\int_{2x}^{3x}(x+{y}^{2})dy\,dx$

Evaluate the iterated integrals.

$\int_{0}^{1}\,\int_{2\sqrt{x}}^{2\sqrt{x}+1}(xy+1)dy\,dx$

Evaluate the iterated integrals.

$\int_{e}^{{e}^{2}}\,\int_{\ln \,u}^{2}(v+\ln \,u)dv\,du$

Evaluate the iterated integrals.

$\int_{1}^{2}\,\int_{-{u}^{2}-1}^{-u}(8uv)dv\,du$

Evaluate the iterated integrals.

$\int_{0}^{1}\,\int_{-\sqrt{1-{y}^{2}}}^{\sqrt{1-{y}^{2}}}(2x+4{x}^{3})dx\,dy$

Evaluate the iterated integrals.

$\int_{0}^{1/2}\,\int_{-\sqrt{1-4{y}^{2}}}^{\sqrt{1-4{y}^{2}}}4dx\,dy$

Let $D$ be the region in the first quadrant bounded by $y=1-{x}^{2},y=4-{x}^{2},$ and the $x$- and $y$-axes.

a. Show that $\iint_{D}x\,dA=\int_{0}^{1}\,\int_{1-{x}^{2}}^{4-{x}^{2}}x\,dy\,dx+\int_{1}^{2}\,\int_{0}^{4-{x}^{2}}x\,dy\,dx$ by dividing the region $D$ into two regions of Type I. b. Evaluate the integral $\iint_{D}x\,dA.$

Let $D$ be the region bounded by $y=1,$ $y=x,$ $y=\ln \,x,$ and the $x$-axis.

a. Show that $\iint_{D}y\,dA=\int_{0}^{1}\,\int_{0}^{x}y\,dy\,dx+\int_{1}^{e}\,\int_{\ln \,x}^{1}y\,dy\,dx$ by dividing $D$ into two regions of Type I. b. Evaluate the integral $\iint_{D}y\,dA.$

a. Show that $\iint_{D}{y}^{2}dA=\int_{-1}^{0}\,\int_{-x}^{2-{x}^{2}}{y}^{2}dy\,dx+\int_{0}^{1}\,\int_{x}^{2-{x}^{2}}{y}^{2}dy\,dx$ by dividing the region $D$ into two regions of Type I, where $D=\{(x,y)|y\ge x,y\ge -x,y\le 2-{x}^{2}\}.$ b. Evaluate the integral $\iint_{D}{y}^{2}dA.$

Let $D$ be the region bounded by $y={x}^{2},y=x+2,$ and $y=-x.$

a. Show that $\iint_{D}x\,dA=\int_{0}^{1}\,\int_{-y}^{\sqrt{y}}x\,dx\,dy+\int_{1}^{4}\,\int_{y-2}^{\sqrt{y}}x\,dx\,dy$ by dividing the region $D$ into two regions of Type II, where $D=\{(x,y)|y\ge {x}^{2},y\ge -x,y\le x+2\}.$ b. Evaluate the integral $\iint_{D}x\,dA.$

The region $D$ bounded by $x=0,y={x}^{5}+1,$ and $y=3-{x}^{2}$ is shown in the following figure. Find the area $A(D)$ of the region $D.$

The region $D$ bounded by $y=\cos \,x,y=4+\cos \,x,$ and $x=\pm \frac{\pi }{3}$ is shown in the following figure. Find the area $A(D)$ of the region $D.$

Evaluate the iterated integrals.

Find the area $A(D)$ of the region $D=\{(x,y)|y\ge 1-{x}^{2},y\le 4-{x}^{2},y\ge 0,x\ge 0\}.$

Let $D$ be the region bounded by $y=1,y=x,y=\ln \,x,$ and the $x$-axis. Find the area $A(D)$ of the region $D.$

Find the average value of the function $f(x,y)=\sin \,y$ on the triangular region with vertices $(0,0),(0,3),$ and $(3,0).$

Find the average value of the function $f(x,y)=-x+1$ on the triangular region with vertices $(0,0),(0,2),$ and $(2,2).$

In the following exercises, change the order of integration and evaluate the integral.

$\int_{-1}^{\pi /2}\,\int_{0}^{x+1}\sin \,x\,dy\,dx$

In the following exercises, change the order of integration and evaluate the integral.

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

In the following exercises, change the order of integration and evaluate the integral.

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

In the following exercises, change the order of integration and evaluate the integral.

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