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## Homework Statement

Use the shell method to find the volume of the solid obtained by rotating the region bounded by y=x

^{2}, y=0 and x=1 about the x-axis.

## Homework Equations

lim Ʃ2∏RhΔw

Δw->0

## The Attempt at a Solution

I realize this is hard to visualize without a graph. I uploaded how I split the graph.

Solved for y and found y=0, 1.

lim Ʃ2∏RhΔw

Δw->0

=lim Ʃ2∏y(1-y)Δy // made it 1-y because y=x2.

Δy->0

=lim Ʃ2∏(y-y

^{2})Δy

Δy->0

=2∏∫y-y

^{2}dy from 0->1

=2∏[y

^{2}/2-y

^{3}/3] from 0->1

=∏/3

Using Fundamental Theorem of Calculus, volume was found to be ∏/3. However this answer is incorrect. The answer is suppose to be ∏/5. I solved the question initially using the washer method and obtained the answer of ∏/5 but this question asks specifically to use shell method. I don't know how I can fix what I did wrong in this question. I assume I set the wrong h value. I also always get confused on what variable to use x or y, when taking into about if the rotation is about the x or y axis.

Any help would be great thanks!