Volume of revolution formula

Rotation 360\degree about the x axis

If we have a curve with the equation y=f(x), we can calculate the volume of the (full) rotation about the x axis using the formula:

V=\pi\int_a^b y^2\space dx

It usually helps to find y^2 first before trying to integrate and find the volume.

Rotation 360\degree about the y axis

If instead we want to rotate the curve of y=f(x) about the y axis by 360 degrees (or 2\pi radians), then we just swap the y and x:

V=\pi\int_a^b x^2\space dx

This does mean you need to rearrange the function of y to make x the subject, then square it.

Rotation of less than 360\degree

If we’re not rotating the full way, then we may want to find a fraction of the full volume. This depends entirely on what the shape looks like. For example, if we already have a full ‘semicircle’ shape in the top two quadrants on the graph, then we’ll have fully created all the volume after just half a revolution about the y-axis (because it’s ‘two sided’ and symmetrical).

The easiest thing to do is think logically about how much it needs to be rotated until it’s created its full volume.

If it needs to rotate 360 degrees to create its full volume, and we’re only rotating it 180 degrees, then we can just take half of the full volume:

V=\frac{1}{2}\pi\int_a^b x^2\space dx

flashcards

QuestionAnswer
Question:What is the formula for the volume of revolution when a curve y=f(x) is rotated 360\degree about the x-axis from x=a to x=b?
Answer:V=\pi\int_a^b y^2\space dx
Question:How should you approach calculating the volume of revolution about the x-axis?
Answer:It usually helps to find y^2 first before trying to integrate and find the volume.
Question:What is the formula for the volume when a curve y=f(x) is rotated 360\degree about the y-axis?
Answer:V=\pi\int_a^b x^2\space dx
Question:What must you do to the function y=f(x) before using the volume of revolution formula about the y-axis?
Answer:You need to rearrange the function of y to make x the subject, then square it.
Question:How do you calculate the volume of revolution for a rotation of less than 360\degree?
Answer:Find a fraction of the full volume based on how much it needs to be rotated to create its full volume; think logically about the shape.
Question:If a ‘semicircle’ shape in the top two quadrants is rotated 180 degrees about the y-axis, why is the full volume created?
Answer:Because it is ‘two sided’ and symmetrical, so only half a revolution is needed.
Question:If a shape requires a 360\degree rotation to create its full volume, but you only rotate it 180\degree, what is the volume?
Answer:V=\frac{1}{2}\pi\int_a^b x^2\space dx