A solid of Revolution

Question

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
y=4x^2, x = 1, y = 0, about the x-axis

Answer 1

The left end point of the interval is not given in the problem.  The right end point is x=1.  

Assuming the left end point is 0.

 

The formula for the volume of revolution of a function between the curve and the x axis about the x axis is V=π∫f(x)² dx.  

 

f(x)=4x² and f(x)²=16x⁴.  So the volume is π∫16x⁴ dx

 

Integral between 0 and 1 yields 16π/5

Answer 2

The volume of a single disk would be πy²dx, not 2πy²dx.

Answer 3

I trust that you can use a graphing calculator or online tool to visualize the parabola y=4x² and the lines x=1 and y=0.

 

Using the disk formula: V=2π∫y²dx over [0,1]

 

Substituting y²=16x⁴ into to the integral, it becomes V=2π∫16x⁴ dx over [0,1]

 

Simplifying and integrating using the power rule, V=32π (1/5) x⁵  over [0,1] = 32π/5.

 

So that's the answer.