# Integrals, Area under the Curve and Solid of Revolution

1.

Evaluate:

∫2cos2 xdx

2.

Figure 12.1

y = 9-x2 , y=5-3x

Sketch the region bounded by the graphs of Figure 12.1, and

then find its area.

3.

Figure 13.1

1?0x4dx

Approximate the integral (Figure 13.1); n=6, by:

a) first applying Simpsonfs Rule and

b) then applying the trapezoidal rule.

4.

Find the mass M (in grams) of a rod coinciding with the interval [0, 4]

which has the density function

(x)= 5 sin 

________________________________________4 x

5.

The region R is bounded by the graphs

x-2y=3 and x=y2

Set up(but do not evaluate) the integral that gives the volume of the solid

obtained by rotating R around the line x = -1.

6.

Figure 16.1

y=2x2

Find the volume of the solid that is generated by rotating the region

formed by the graphs of Figure 16.1 and y = 4x about the line x = 3.

7.

Use the method of cylindrical shells to find the volume of the solid

rotated about the line x = -1 given the conditions:

y = x3 - x2; y = 0; x = 0

8.

Find the length of the graph of

y = 1

________________________________________3 x3/2 - x1/2

from

(1, - 2

________________________________________3 )

to

(4, 2

________________________________________3 )

9.

A 10-ft trough filled with water has a semicircular cross section of

diameter 4 ft. How much work is done in pumping all the water over

the edge of the trough? Assume that water weighs

62.5lb/ft3

10.

Figure 20.1

y = x2

________________________________________2

The region in the first quadrant bounded by the graphs of y = x and Figure 20.1 is rotated around the line y = x. Find

(a) the centroid of the region and

(b) the volume of the solid of revolution.

#### Solution Summary

Integrals, Area under the Curve and Solid of Revolution are investigated.