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11 votes
11 votes
Find the volume of the solid formed by revolving the region bounded by y = (x - 2)² and y = x about the y-axis.​

User Jonathan Corwin
by
2.6k points

1 Answer

11 votes
11 votes

Answer:

22.51 unit^3 to nearest hundredth.

Explanation:

First find the points of intersection of the 2 functions:

y = (x - 2)^2

y = x

So x = (x - 2)^2

x^2 - 4x + 4 - x = 0

x^2 - 5x + 4 = 0

(x - 1)( x - 4) = 0

So they intersect at (1, 1) and (4,4).

y = (x - 2)^2

----> x - 2 = √y

----> x = √y + 2

Required volume

4

= ∫π (√y + 2)^2 dy - ∫π y^2 dy

1

4

= π ∫( y + 2√y + 4) dy - π ∫ y^2 dy

1

4

= π [ y^2/2 + 2y^3/2 / 3/2 + 4y] - [π[y^3/3]

1

= π [8 + 10.666 + 16] - [0.5 + 2 + 4} - π[21.333- 0.333]

= 88.486 - 65.973

= 22.513.

User Robert Nubel
by
3.1k points
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