일반 영역의 이중적분

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

Find the volume when you revolve the region around the $x\text{-axis.}$

Find the volume when you revolve the region around the $y\text{-axis.}$

An integral can be interpreted as giving the signed area over an interval; a double integral can be interpreted as giving the signed _____ over a region.

Explain why the following statement is false: “Fubini's Theorem states that $\displaystyle \int_a^b\int_{g_1(x)}^{g_2(x)} f(x,y)\ dy\ dx = \int_a^b\int_{g_1(y)}^{g_2(y)} f(x,y)\ dx\ dy$.”

Explain why if $f(x,y)>0$ over a region $R$, then

$\iint_Rf(x,y)\ dA>0$.

If $\iint_R f(x,y)\ dA = \iint_R g(x,y)\ dA$, does this imply $f(x,y) = g(x,y)$?

In Exercises 5–10,

(a) Evaluate the given iterated integral, and (b) rewrite the integral using the other order of integration.

$\displaystyle \int_1^2\int_{-1}^1\left(\frac xy+3\right)\ dx\ dy$

In Exercises 5–10,

(a) Evaluate the given iterated integral, and (b) rewrite the integral using the other order of integration.

$\displaystyle \int_{-\pi/2}^{\pi/2}\int_{0}^\pi\left(\sin x\cos y\right)\ dx\ dy$

In Exercises 5–10,

(a) Evaluate the given iterated integral, and (b) rewrite the integral using the other order of integration.

$\displaystyle \int_{0}^{4}\int_{0}^{-x/2+2}\left(3x^2-y+2\right)\ dy\ dx$

In Exercises 5–10,

(a) Evaluate the given iterated integral, and (b) rewrite the integral using the other order of integration.

$\displaystyle \int_{1}^{3}\int_{y}^{3}\left(x^2y-xy^2\right)\ dx\ dy$

In Exercises 5–10,

(a) Evaluate the given iterated integral, and (b) rewrite the integral using the other order of integration.

$\displaystyle \int_{0}^{1}\int_{-\sqrt{1-y}}^{\sqrt{1-y}}\left(x+y+2\right)\ dx\ dy$

In Exercises 5–10,

(a) Evaluate the given iterated integral, and (b) rewrite the integral using the other order of integration.

$\displaystyle \int_{0}^{9}\int_{y/3}^{\sqrt{y}}\left(xy^2\right)\ dx\ dy$

In Exercises 11–18:

(a) Sketch the region $R$ given by the problem. (b) Set up the iterated integrals, in both orders, that evaluate the given double integral for the described region $R$. (c) Evaluate one of the iterated integrals to find the signed volume under the surface $z=f(x,y)$ over the region $R$.

$\displaystyle \iint_R x^2y\ dA$, where $R$ is bounded by $y=\sqrt{x}$ and $y=x^2$.