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## Main Question or Discussion Point

**How do I calculate the volume of this??**

What is the volume of the region bounded by y = x^2, y = 1, and the y-axis rotated around the y -axis

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What is the volume of the region bounded by y = x^2, y = 1, and the y-axis rotated around the y -axis

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mathwonk

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I should do this using integral.. I was thinking of slicing this region vertically but then how do I represent this in integrals?? I have to take integrals from -1 to 1 right?

I would evaluate this as the integral from 0 to 1 of (1-squareroot of y)^2 dy

is that right??

I would evaluate this as the integral from 0 to 1 of (1-squareroot of y)^2 dy

is that right??

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HallsofIvy

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horizontallywill be disks with radius x. Each would have area [itex]\pi x^2= \pi y[/itex] and each infinitesmal disc will have volume [itex]\pi y dy[/itex]. Integrate that.

I am sorry that's my mistake. I would slice it horizontally and take the integral from 0 to 1. And shouldn't the radius be 1-square root of y?? Because it's the intersection of y = x^2 and y = x

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HallsofIvy

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Where did you get y= x from? Your original post was:I am sorry that's my mistake. I would slice it horizontally and take the integral from 0 to 1. And shouldn't the radius be 1-square root of y?? Because it's the intersection of y = x^2 and y = x

Every horizontal "slice" is a circle with center at the y-axis, x= 0, and the end of a radius at [itex]x= \sqrt{y}[/itex]. Of course, the area of the disk is [itex]\pi x^2= \pi y[/itex].What is the volume of the region bounded by y = x^2, y = 1, and the y-axis rotated around the y -axis