$y=e^{x}, x=0, y=4$

$y=x^{2}, y=18-x^{2}$

$x=2y^{2}, x=2$

$y=\dfrac{1}{x},$ $x+$ $y=4$

$y=4-x^{2},$ $y=3x$

$y=x^{2},$ $\ y=4x-x^{2}$; about the $x$-axis

$y=x^{2}-5x+6, y=0$; about the $y$-axis

$x=y^{2}-4, \ x=0$; about the $y$-axis

$xy=1$, $x=1$, $x=2$, $y=0$; about the $x$-axis

$y=x^{2}-4,$ $y=0;$ about the line $y=-4$

$y=4x-x^{2}$ and the $x$-axis; about the $x$-axis

$y^{2}=8x$, $y\geq 0$, and $x=2;$ about the line $x=2$

$y=\dfrac{x^{3}}{2}$, $y=0$, $x=2;$ about the $y$-axis

$y=e^{x}$, $y=1$, $x=1;$ about the $x$-axis

$y^{2}=x^{3}$, $y=8$, $x=0$; about the line $x=4$

$y=x^{3/2}+4$ from $x=2$ to $x=5$

$y=\dfrac{x^{3}}{6}+\dfrac{1}{2x}$ from $x=2$ to $x=6$

$2y^{3}=x^{2}$ from $y=0$ to $y=2$

Center of Mass Find the center of mass of the system of masses: $m_{1}=1, m_{2}=3, m_{3}=8, m_{4}=1$ located, respectively, at $-1, 2, 14, \textrm{ and } 0$.

Moments and Center of Mass Find the moments $M_{x}$ and $M_{y}$ and the center of mass of the system of masses: $m_{1}=2, m_{2}=2, m_{3}=3, m_{4}=2$ located, respectively, at the points $(-4,4), (2,3), (4,4), (-3,-5)$.

Area of a Triangle Use integration to find the area of the triangle formed by the lines $x-y+1=0$, $3x+y-13=0$, and $x+3y-7=0$.

Volume A solid has a circular base of radius $4$ units. Slices taken perpendicular to a fixed diameter are equilateral triangles. Find its volume.

Volume Find the volume of the solid generated by revolving the region bounded by the graph of $4x^2+9y^2=36$ in the first quadrant about the $x$-axis.

Volume of a Cone Find the volume of an elliptical cone with base $\dfrac{x^{2}}{4}+y^{2}=1$ and height $5$. {Hint: The area $A$ of an ellipse with semi-axes $a$ and $b$ is $A=\pi ab$.)

Volume The base of a solid is enclosed by $4x+5y=20$, $x=0$, $y=0$. Every cross section perpendicular to the base along the $x$-axis is a semicircle.Find the volume of the solid.

Arc Length Find the point $P$ on $y=\dfrac{2}{3}x^{3/2}$ to the right of the $y$-axis so that the length of the curve from $(0,0)$ to $P$ is $\dfrac{52}{3}$.

Work Find the work done in raising an 800-$\textrm{ lb}$ anchor $150\textrm{ ft}$ with a chain weighing $20\textrm{ lb/ft}$.

Work Find the work done in raising a container of $1000\textrm{ kg}$ of silver ore from a mine $1200\textrm{ m}$ deep with a cable weighing $3\textrm{ kg/m}$.

Work Pumping Water A hemispherical water tank has a diameter of $12{m}$. It is filled to a depth of $4{m}$. How much work is done in pumping all the water over the edge? (Use $\rho =1000\textrm{ kg/m}^{3}$.)

Work of a Spring A spring with an unstretched length of $0.6 \textrm{ m}$ requires a force of $4\textrm{ N}$ to stretch it to $0.8\textrm{ m}$. How much work is done in stretching it to $1.4\textrm{ m}$?

Work of a Spring Find the unstretched length of a spring if the work required to stretch the spring from $1.0$ to $1.4\textrm{ m}$ is half the work required to stretch it from $1.2$ to $1.8{\text{m}}$.

Hydrostatic Force A trough of trapezoidal cross section is $2$ ft wide at the bottom, $4$ ft wide at the top, and $3$ ft deep. What is the force due to liquid pressure on the end, if it is full of water?

Hydrostatic Force A cylindrical tank is on its side. It has a diameter of $10{m}$ and is full to a depth of $5\text{m}$ with gasoline that has a density of $737\text{kg/m}^{3}$. What is the force due to liquid pressure on the end?

Hydrostatic Force A dam is built in the shape of a trapezoid $1000{ft}$ long at the top, $700{ft}$ long at the bottom, and $80\textrm{ ft}$ deep. Determine the force of water on the dam if:

Centroid Find the centroid of the lamina bounded by the graphs of $y=\sqrt{x}, y=0$, and $x=9$.

Pappus Theorem for Volume Find the volume of the torus with an outer diameter of $5\text{cm}$ and an inner diameter of $2\text{cm}$.

Volume of a Solid of Revolution Find the volume generated when the triangular region bounded by the lines $x=3, y=0$, and $2x+y-12=0$ is revolved about the $y$-axis.

Volume of a Solid of Revolution Find the volume generated when the region bounded by the graphs of $y=3 \sqrt{x}$ and $y=-x^{2}+6x-2$ is revolved about each of the following lines: