Question 04 (b)
Consider the function for 0 ≤ x ≤ 1.
Find the volume of the bowl when rotating around the -axis. (Note: for this problem there are two ways to set the relevant integral up - either one is fine. One way involves rewriting as .)
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.
Draw a picture. Visualize the two ways of integrating as referred to in the hint.
Write down two integrals that represent the volume of the solid. Which one would you prefer to evaluate?
In this case, both integrals are possible to evaluate using the techniques that you've learned. It is a good reality check (and also a good exercise) to compute both integrals and ensure that you obtain the same result.
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
|-axis. Each slice has volume , where is its radius and is its thickness. The bowl will be made by rotating the shaded blue region in the 2D figure , then its radius is equal to , where . Hence, In the limit
Disc Method. In this solution, we consider the volume as a sum of slices, cut orthogonal to the
Note that the limits of integration are and since ranges from 0 to 1.
We use integration by parts to evaluate this integral, letting So in order to perform the integration by parts we need to recall the antiderivative of . To calculate this antiderivative we again use integration by parts:
We now use this antiderivative to perform the integration by parts in the integral for Hence,
| where is its radius, is its height and is its thickness. The bowl will be made up by rotating the blue shaded region in the 2D figure , and . In the limit we obtain the volume by integrating along the x-axis:
Shell method. In this solution, we consider the solid to be a sum of cylindrical shells. Each cylindrical shell has volume
(We used integration by parts in the second integral to get from the third to the fourth line.)Reality check: Note that we obtain the same value for with both methods. To visualize the full bowl we can look at the 3D figure