zakownikbj

zakownikbj

Answered

2022-09-23

Finding volume of solid
Suppose that a solid is formed in such a way that each cross section perpendicular to the x-axis, for 0 x 1, is a disk, a diameter of which goes from the x-axis out to the curve y = x .
Find the volume of the solid.
For this I use the disk formula. So π 0 1 ( x ) 2 d x . .
When I do this, I get π 5 . The answer is π 8 . What am I doing wrong?

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Answer & Explanation

Klecanlh

Klecanlh

Expert

2022-09-24Added 11 answers

Explanation:
You are told that the cross-sections perpendicular to the x-axis are disks where the diameter is from y = x to y = 0.
- The diameter of this disk is x
- The radius of this disk is x / 2
- The cross-sectional area is π ( x / 2 ) 2
- V = "Cross-sectional Area"   d  "axis perp to cross section"
V = π 0 1 ( x 2 ) 2 d x = π 0 1 x 4 d x = π x 2 8 | 0 1 = π 8

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Dymnembalmese2n

Dymnembalmese2n

Expert

2022-09-25Added 2 answers

Step 1
You're doing two things wrong.
One, it is explicitly saying that the diameter of each shell is x , so the radius would be x / 2
Step 2
Next, you're integrating wrong. The integral you have ( π 0 1 ( x ) 2 d x) should give π / 2 which, multiplied by the new 1 4 from above, gives the correct answer.

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