Rotate the area between the curve y=x^2 and y=9, bounded in the first quadrant, around the vertical line x=3. Then find the height (m) of the horizontal line that divides the resulting volume in half.

Jonas Huff

Jonas Huff

Answered question

2022-11-03

Check my work? Finding line that divides rotational volume into equal parts. Not getting right answer.
I need to rotate the area between the curve y = x 2 and y = 9, bounded in the first quadrant, around the vertical line x = 3. I then must find the height (m) of the horizontal line that divides the resulting volume in half.
I've been trying to set up two integrals with washers. One from m 9 3 2 ( 3 x ) 2 d y and the other for the bottom region 0 m 3 2 ( 3 x ) 2 d y. The 3 is the outer radius of the washer and the 3 x gives the inner hole of the washer. I can't seem to get the right answer.
π m 9 ( 6 y y ) d y = π ( 4 y 3 2 1 2 y 2 ) evaluated from 9 to m = π ( 135 2 ( 4 m 3 2 1 2 m 2 ). Similarly, for the bottom region integral, I get π ( 4 m 3 2 1 2 m 2 ). I then try to set the volumes equal to each other and solve for m.
I believe the answer I should get is 9 2 3 but I don't get that value for m.
If I continue, I get 135 2 = 8 m 3 2 - m 2 but I don't see any easy way to solve without using a calculator.

Answer & Explanation

sellk9o

sellk9o

Beginner2022-11-04Added 11 answers

Step 1
The volume of a rotational solid is π r 2 d x, where x is an arbitrary variable and r is the radius in terms of x. Now, somewhat counterintuitively, let's put r in terms of y. For any value of y R + , x = y . The radius of the solid is the distance between x = 3 and f ( y ) = y , which is given by 3 y .
Step 2
Thus, we have V = π 0 9 ( 3 y ) 2 d y = π ( 0 9 ( 9 + y ) d y 0 9 6 y d y ) = π ( 9 y + 1 2 y 2 | 0 9 4 y 3 / 2 | 0 9 ) = π ( 81 + 40 1 2 108 ) = 13.5 π
The reason you're off is, as far as I can tell, you're still rotating around x = 0 instead of rotating about x = 3, as the question dictates

Do you have a similar question?

Recalculate according to your conditions!

New Questions in High school geometry

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?