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Integrals, Area under the Curve and Solid of Revolution

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∫2cos2 xdx

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.
Figure 13.1

Approximate the integral (Figure 13.1); n=6, by:
a) first applying Simpsonfs Rule and
b) then applying the trapezoidal rule.
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

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.

Figure 16.1

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.
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

Find the length of the graph of
y = 1
________________________________________3 x3/2 - x1/2
(1, - 2
________________________________________3 )
(4, 2
________________________________________3 )

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

Figure 20.1
y = x2

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.

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Solution Summary

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

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