Question

Calculate the volume of the solid generated by revolving the region bounded by the curve and the line around the x-axis.

Ask by Wheeler Owen. in Netherlands
Jan 05,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The volume of the solid is .

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Beyond the Answer

To find the volume of the solid generated by revolving the region bounded by the curve and the line around the x-axis, we can use the disk method. First, we determine the points of intersection of the curve and the line.
Setting equal to :
Thus, the solutions are and .
The volume of the solid of revolution is given by the integral:
where is the distance from the x-axis to the curve, which in this case would be the line minus .
Thus, .
Now, we plug this into our volume formula:
Expanding the integrand:
So we need to compute:
We’ll evaluate the integral term by term:
  1. Since is an even function,
  2. For ,
Combining these results, we find:
Finding a common denominator (which is 15):
So,
Thus, the final volume is:

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy