Geometric and physical applications of double integrals

Find the area bounded by the following curves, using the formula $S=\displaystyle \iint_{S} dx\, dy$: $xy=a^{2}$, $x+y=\dfrac {5a}{2}$ $(a>0)$.

Fig. 8.34

Find the area bounded by the following curves, using the formula $S=\displaystyle \iint_{S} dx\, dy$: $y^{2}=2px+p^{2}$, $y^{2}=-2qx+q^{2}$ $(p>0,q>0)$.

Fig. 8.35

Find the area bounded by the following curves, using the formula $S=\displaystyle \iint_{S} dx\, dy$: $(x-y)^{2}+x^{2}=a^{2}$ $(a>0)$.

Fig. 8.36

Sketch a solid whose volume equals the integral

Sketch the volumes represented by the following double integrals:

Find the volume bounded by the following surfaces: $z=1+x+y$, $z=0$, $x+y=1$, $x=0$, $y=0$.

Find the volume bounded by the following surfaces: $x+y+z=a$, $x^{2}+y^{2}=R^{2}$, $x=0$, $y=0$, $z=0$ $(a\geqslant R\sqrt{2})$.

Find the volume bounded by the following surfaces: $z=x^{2}+y^{2}$, $y=x^{2}$, $y=1$, $z=0$.

Find the volume bounded by the following surfaces: $z=\cos x\cos y$, $z=0$, $\left|x+y\right|\leqslant \dfrac {\pi }{2}$, $\left|x-y\right|\leqslant \dfrac {\pi }{2}$.

Find the volume bounded by the following surfaces: $z=\sin \dfrac {\pi y}{2x}$, $z=0$, $y=x$, $y=0$, $x=\pi$.

Find the volume bounded by the following surfaces: $z=xy$, $x+y+z=1$, $z=0$.

Find the mass of a square plate of side $a$, given that the density at each point of the plate is proportional to the distance from that point to one of the square's vertices, and equals $\rho_{0}$ at the center of the square.

Find the coordinates of the centroid of the uniform lamina bounded by the following curves: $ay=x^{2}$, $x+y=2a$ $(a>0)$.

Find the coordinates of the centroid of the uniform lamina bounded by the following curves: $\sqrt{x}+\sqrt{y}=\sqrt{a}$, $x=0$, $y=0$.

Find the coordinates of the centroid of the uniform lamina bounded by the following curves: $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ $(x\geqslant 0,y\geqslant 0)$.

Find the coordinates of the centroid of the uniform lamina bounded by the following curves: $\left(\dfrac {x}{a}+\dfrac {y}{b}\right)^{3}=\dfrac {xy}{c^{2}}$ (a loop).

Find the coordinates of the centroid of the uniform thin plate bounded by the following curves: $x=a(t-\sin t)$, $y=a(1-\cos t)$ $(0\leqslant t\leqslant 2\pi )$, $y=0$.

Find the moments of inertia $I_{x}$ and $I_{y}$ about the coordinate axes $Ox$ and $Oy$, taking $\rho =1$, of the area bounded by the following curves: $\dfrac {x}{b_{1}}+\dfrac {y}{h}=1$, $\dfrac {x}{b_{2}}+\dfrac {y}{h}=1$, $y=0$ $(b_{1}>0,b_{2}>0,h>0)$.

Find the moments of inertia $I_{x}$ and $I_{y}$ about the coordinate axes $Ox$ and $Oy$, taking $\rho =1$, of the area bounded by the following curves: $(x-a)^{2}+(y-a)^{2}=a^{2}$, $x=0$, $y=0$ $(0\leqslant x\leqslant a)$.

On an elliptical region

the pressure distribution acting on a body is given by the formula

Find the average pressure on the body over this region.

A meadow has the shape of a rectangle with sides $a$ and $b$, uniformly covered with cut grass of density $p$ kilograms per square meter. Suppose the work needed to carry a weight of $P$ kilograms a distance $r$ away is $kPr$ $(0\leqslant k\leqslant 1)$. To gather all the hay at the center of the meadow, what is the minimum amount of work that must be done?

Find the volume bounded by the surfaces $x+y+z=6$, $x=0$, $z=0$, $x+2y=4$.

Find the volume bounded by the surfaces $x-y+z=6$, $x+y=2$, $x=y$, $y=0$, $z=0$.

Find the volume bounded by the surfaces $x+y+z=6$, $x=y$, $y=0$, $x=3$, $z=0$.

Find the volume bounded by the surfaces $x+y+z=a$, $3x+y=a$, $3x+2y=2a$, $y=0$, $z=0$.

Prove that the volume bounded by the plane $z=mx+ny+p$ and the paraboloid $z=x^{2}+y^{2}$ equals half the area of the circle $x^{2}+y^{2}=mx+cy+p$, multiplied by the difference of the quantities $x^{2}+y^{2}$ and $mx+cy+p$, taken at the center of that circle.

Prove that the volume of the segment cut off by a plane from an elliptic paraboloid equals the area of the base of the segment, multiplied by half the height of the segment.

Let a set $G$ be elementary with respect to the $O z$ axis, i.e.,

where $\Omega$ is a closed measurable set in $R^{2}$, and the functions $\varphi$ and $\psi$ are continuous on $\Omega$. Prove that the formula for the volume of $G$ holds:

In particular, if $\psi (x ; y) \geqslant 0$, and $\varphi (x ; y) = 0$ on $\Omega$ (i.e., $\Omega$ is the base of the cylindrical set $G$, and the graph of $\psi$ is the "lid"), then

¹Find the area of the domain bounded by the curves:

Find the volumes of the solids bounded by the surfaces (see formulas (24), (25)).

Find the volumes of the solids bounded by the surfaces (see formulas (24), (25)).