# Calculus

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The region between the graphs of x=y^2 and x=4y is rotated around the line y=4.
The volume of the resulting solid is?

• Calculus -

I assume you made a sketch.
You should have the two graphs intersecting at (0.0) and (16,4)

So we will washer discs with outer ratius of 4 - x/4 and inner radius of 4 - √x

area of washer disc = π(4-x/4)^2 = π(4-√x)^2
= π(16 - 2x + x^2/4 - (16 - 8√x + x)
= π( -3x + x^2 + 8x^(1/2)

volume = π∫(-3x + x^2 + 8x^(1/2) dx from 0 to 16
= π [ (-3/2)x^2 + (1/3)x^3 + (16/3)x^(3/2) ] from 0 to 16

= π(-384 + 4096/3 + 1024/3 - 0]
= π(3968/3)

check my arithmetic

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