Volume of Revolution

In this page, we will learn about how to find the volume generated by rotating a region around the x axis and the y axis. The formule for these are not difficult to use. The difficulty often comes with applying the integration techniques. It is therefore recommended that you revise these techniques before you go through this topic. In particular, you should be confident with Integration by Substitution and  Integration by Parts.


Key Concepts

On this page, you should learn about

  • finding the volume generated by rotating a region under a curve about the x axis/y axis
  • finding the volume generated by rotating a region bounded by two graphs about the x axis/y axis

Summary

Test Yourself

Here is a quiz that practises the skills from this page



START QUIZ!

Exam-style Questions

Question 1

a) Show that \(\large \int {x^2e^{2x}\,dx}=\frac{1}{4}e^{2x}(2x^2-2x+1)+c\)

The following graph shows the function \(\large f(x)=xe^x\)

b) Show that the equation of the tangent to the graph at x = 1 has the equation \(\large y=(2e)x-e\)

The region bounded by \(\large f\), the tangent \(\large y=(2e)x-e\) and y = 0 is rotated \(\large 2\pi\) around the x axis

c) Find the volume of this solid in terms of \(\large \pi\)

Hint

Full Solution

Question 2

The following graph shows the curve defined by the equation \(\large (x-1)^2+y^2=4\).

The region bounded by the curve and the lines y = 0 and x = 0 is rotated \(\large 2\pi\) around the x axis.

Find the volume of this solid in terms of \(\large \pi\)

Hint

Full Solution

 

Question 3

The graph shows \(\large f(x) = -\arctan(1-x^2)\), the tangent to the curve at (1 , 0) and the tangent to the curve at the point \(\large(-\frac{\pi}{4},0)\).

The shaded region is bounded by the curve and the two tangents. This region is rotated \(\large 2\pi\) around the y axis to forma solid.

Find the volume of this solid correct to 3 significant figures.

Hint

Full Solution

 

Question 4

The following diagram shows the graph of \(\large x^2=\cos^3y\) for \(\large -\frac{\pi}{2}\le y\le\frac{\pi}{2}\)

The shaded region R is the area bounded by the curve, the y axis and the lines \(\large y=-\frac{\pi}{2}\) and \(\large y=\frac{\pi}{2}\).

The rgion is rotated about the y axis through \(\large 2\pi\) to form a solid.

Show that the volume of the solid is \(\large \frac{4\pi}{3}\)

Hint

Full Solution

Question 5

The following diagram shows the graph of the function \(\large f(x)=\frac{\sqrt{x}}{\cos x}\) for \(\large 0\le x \le\frac{\pi}{2}\)

The shaded region is the area bounded by f, the x axis, x= 0 and x = \(\large \frac{\pi}{3}\)

The region is rotated about the x axis through \(\large 2\pi\) to form a solid.

Find the volume of the solid.

Hint

Full Solution

 

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