A draftsman is asked to determine the amount of material required to produce a machine part. The diameters d of the part at equally spaced points x are listed in the table. The measurements are listed in centimeters.

x	0	1	2	3	4	5
d	4.2	3.8	4.2	4.7	5.2	5.7

x	6	7	8	9	10
d	5.8	5.4	4.9	4.4	4.6

Use these data with Simpson's Rule to approximate the volume of the part.

The graphs of y=x$^4$-2x$^2$+1 and y=1-x$^2$ intersect at three points. However, the area between the curves can be found by a single integral. Explain why this is so, and write an integral for this area.

A tank on a water tower is a sphere of radius 50 feet. Determine the depths of the water when the tank is filled to one-fourth and three-fourths of its total capacity. (Note: Use the root-finding capabilities of a graphing utility after evaluating the definite integral.)

The arc of y=4-(x$^2$/4) on the interval [0,4] is revolved about the line y=b.

Use calculus to find the value of b that minimizes the volume of the solid, and compare the result with the answer to part (b).

Use a graphing utility to graph the function in part (a), and use the graph to approximate the value of b that minimizes the volume of the solid.

The arc of y=4-(x$^2$/4) on the interval [0,4] is revolved about the line y=b.

Find the volume of the resulting solid as a function of b.

Find the centroid of the region bounded by the graphs of the equations.

$$y=x^{2 / 3}, y=\frac{1}{2} x$$

Set up and evaluate the definite integral that gives the area of the region bounded by the graph of the function f(x)=1 / x$^2$+1,(1, 1/2) and the tangent line to the graph at the indicated point.

In the following exercise, set up and evaluate the integral that gives the volume of the solid formed by revolving the region about the x-axis.

$$y=x^2, y=x^3$$

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the y-axis.

5y=x$^2$, y=0, x=2

Find the arc length of the graph of the function over the indicated interval.

y=3/2 x$^{2 / 3}$, [1,8]

Find the fluid force on the vertical side of the tank, where the dimensions of rectangle are 4 $\times$ 3 in feet. Assume that the tank is full of water.

Set up the definite integral that gives the area of the region.

f(x)=3(x$^3$-x)

g(x)=0

Sketch the region bounded by the graphs of the equations and determine the area of the region.

y=x, $y=x^3$