다중적분 전체 문제

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.

For the following problems, find the center of mass of the region.

$\rho (x,y)=(y+1)\sqrt{x}$ in the region bounded by $y={e}^{x},$ $y=0,$ and $x=1.$

$\rho (x,y,z)=z$ on the inverted cone with radius $2$ and height $2.$

The volume an ice cream cone that is given by the solid above $z=\sqrt{({x}^{2}+{y}^{2})}$ and below ${z}^{2}+{x}^{2}+{y}^{2}=z.$

If the compacted trash used to build Mount Holly on average has a density $400{\,\text{lb/ft}}^{3},$ find the amount of work required to build the mountain.

In reality, it is very likely that the trash at the bottom of Mount Holly has become more compacted with all the weight of the above trash. Consider a density function with respect to height: the density at the top of the mountain is still density $400{\,\text{lb/ft}}^{3}$ and the density increases. Every $100$ feet deeper, the density doubles. What is the total weight of Mount Holly?

[T] The temperature of Earth’s layers is exhibited in the table below. Use your calculator to fit a polynomial of degree $3$ to the temperature along the radius of the Earth. Then find the average temperature of Earth. (Hint: begin at $0$ in the inner core and increase outward toward the surface)

Source: http://www.enchantedlearning.com/subjects/astronomy/planets/earth/Inside.shtml

[T] The density of Earth’s layers is displayed in the table below. Using your calculator or a computer program, find the best-fit quadratic equation to the density. Using this equation, find the total mass of Earth.

Source: http://hyperphysics.phy-astr.gsu.edu/hbase/geophys/earthstruct.html