다중적분 전체 문제

In Exercises 15–16, special double integrals are presented that are especially well suited for evaluation in polar coordinates.

Consider $\displaystyle \iint_R e^{-(x^2+y^2)}\ dA$.

(a) Why is this integral difficult to evaluate in rectangular coordinates, regardless of the region $R$? (b) Let $R$ be the region bounded by the circle of radius $a$ centered at the origin. Evaluate the double integral using polar coordinates. (c) Take the limit of your answer from (b), as $a\to\infty$. What does this imply about the volume under the surface of $\displaystyle e^{-(x^2+y^2)}$ over the entire $x$-$y$ plane?

In Exercises 15–16, special double integrals are presented that are especially well suited for evaluation in polar coordinates.

The surface of a right circular cone with height $h$ and base radius $a$ can be described by the equation $\displaystyle f(x,y) = h-h\sqrt{\frac{x^2}{a^2}+\frac{y^2}{a^2}}$, where the tip of the cone lies at $(0,0,h)$ and the circular base lies in the $x$-$y$ plane, centered at the origin.

Confirm that the volume of a right circular cone with height $h$ and base radius $a$ is $\displaystyle V = \frac13\pi a^2h$ by evaluating $\displaystyle \iint_R f(x,y)\ dA$ in polar coordinates.

Why is it easy to use “mass” and “weight” interchangeably, even though they are different measures?

Given a point $(x,y)$, the value of $x$ is a measure of distance from the _____-axis.

We can think of $\iint_R\ dm$ as meaning “sum up lots of _____”

What is a “discrete planar system?”

Why does $M_x$ use $\iint_R y\delta(x,y)\ dA$ instead of $\iint_R x\delta(x,y)\ dA$; that is, why do we use “$y$” and not “$x$”?

In Exercises 7–10, point masses are given along a line or in the plane. Find the center of mass $\overline{x}$ or $(\overline{x},\overline{y})$, as appropriate. (All masses are in grams and distances are in cm.)

$m_1=4$ at $x=1$; $m_2=3$ at $x=3$; $m_3=5$ at $x=10$

In Exercises 7–10, point masses are given along a line or in the plane. Find the center of mass $\overline{x}$ or $(\overline{x},\overline{y})$, as appropriate. (All masses are in grams and distances are in cm.)

$m_1=2$ at $x=-3$; $m_2=2$ at $x=-1$;

$m_3=3$ at $x=0$; $m_4=3$ at $x=7$

In Exercises 7–10, point masses are given along a line or in the plane. Find the center of mass $\overline{x}$ or $(\overline{x},\overline{y})$, as appropriate. (All masses are in grams and distances are in cm.)

$m_1=2$ at $(-2,-2)$; $m_2=2$ at $(2,-2)$;

$m_3=20$ at $(0,4)$

In Exercises 7–10, point masses are given along a line or in the plane. Find the center of mass $\overline{x}$ or $(\overline{x},\overline{y})$, as appropriate. (All masses are in grams and distances are in cm.)

$m_1=1$ at $(-1,-1)$; $m_2=2$ at $(-1,1)$;

$m_3=2$ at $(1,1)$; $m_4=1$ at $(1,-1)$

In Exercises 11–18, find the mass/weight of the lamina described by the region $R$ in the plane and its density function $\delta(x,y)$.

$R$ is the rectangle with corners $(1,-3)$, $(1,2)$, $(7,2)$ and $(7,-3)$; $\delta(x,y) = 5$gm/cm$^2$

In Exercises 11–18, find the mass/weight of the lamina described by the region $R$ in the plane and its density function $\delta(x,y)$.

$R$ is the rectangle with corners $(1,-3)$, $(1,2)$, $(7,2)$ and $(7,-3)$; $\delta(x,y) = (x+y^2)$gm/cm$^2$

In Exercises 11–18, find the mass/weight of the lamina described by the region $R$ in the plane and its density function $\delta(x,y)$.

$R$ is the triangle with corners $(-1,0)$, $(1,0)$, and $(0,1)$; $\delta(x,y) = 2$lb/in$^2$

In Exercises 11–18, find the mass/weight of the lamina described by the region $R$ in the plane and its density function $\delta(x,y)$.

$R$ is the triangle with corners $(0,0)$, $(1,0)$, and $(0,1)$; $\delta(x,y) = (x^2+y^2+1)$lb/in$^2$

In Exercises 11–18, find the mass/weight of the lamina described by the region $R$ in the plane and its density function $\delta(x,y)$.

$R$ is the disk centered at the origin with radius 2; $\delta(x,y) = (x+y+4)$kg/m$^2$

In Exercises 11–18, find the mass/weight of the lamina described by the region $R$ in the plane and its density function $\delta(x,y)$.

$R$ is the circle sector bounded by $x^2+y^2=25$ in the first quadrant; $\delta(x,y) = (\sqrt{x^2+y^2}+1)$kg/m$^2$

In Exercises 11–18, find the mass/weight of the lamina described by the region $R$ in the plane and its density function $\delta(x,y)$.

$R$ is the annulus in the first and second quadrants bounded by $x^2+y^2=9$ and $x^2+y^2=36$; $\delta(x,y) = 4$lb/ft$^2$

In Exercises 11–18, find the mass/weight of the lamina described by the region $R$ in the plane and its density function $\delta(x,y)$.

$R$ is the annulus in the first and second quadrants bounded by $x^2+y^2=9$ and $x^2+y^2=36$; $\delta(x,y) = \sqrt{x^2+y^2}$lb/ft$^2$

The moment of inertia $I$ is a measure of the tendency of a lamina to resist rotating about an axis or continue to rotate about an axis. $I_x$ is the moment of inertia about the $x$-axis, $I_y$ is the moment of inertia about the $x$-axis, and $I_O$ is the moment of inertia about the origin. These are computed as follows:

• $\displaystyle I_x = \iint_R y^2\ dm$
• $\displaystyle I_y = \iint_R x^2\ dm$
• $\displaystyle I_O = \iint_R \big(x^2+y^2\big)\ dm$ In Exercises 27–30, a lamina corresponding to a planar region $R$ is given with a mass of 16 units. For each, compute $I_x$, $I_y$ and $I_O$.

$R$ is the $4\times 4$ square with corners at $(-2,-2)$ and $(2,2)$ with density $\delta(x,y) = 1$.