volume of a solid

Question

Find the volume of the solid formed by revolving the region bounded by the graphs of y = x2, x = 3, and y = 4 about the y-axis. (4 points)

25pi/3

97pi/3

173pi/3
None of these

Answer 1

The best way to find the volume is to use the shell method.

 

The region you are rotating lies between the three curves given.

 

The shell radius is x, since a vertical strip in the region is being rotated about the y-axis and that is how far the strip is from the y-axis.

 

The shell height is x² - 4, since the region resembles a triangle with top curve the parabola and bottom curve the horizontal line y = 4.

 

To find the volume, you use the shell method formula:

 

Area = 2π ∫ (shell radius)(shell height) dx, with the limits of integration x = 2 to x = 3. (The x = 2 comes from the point of intersection of the line y = 4 with the parabola at (2,4).)

 

When you set this up, you will get the definite integral

 

Area = 2π ∫₂³ x(x² - 4) dx = 2π ∫₂³ (x³ - 4x) dx = 2π [x⁴/4 - 2x²]₂³ = 2π [81/4 - 18 - (4 - 8)] = 2π (81 /4 - 14)

 

        = 2π (25/4) = 25π/2 .

 

You should check my work because I did this without using pencil and paper and while typing the answer at the same time (which is not easy).