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A region is bounded by x=y^2 and x=4 and y=0 and revolved about the line x=5. Find the volume using shell method.

User Ullan
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1 Answer

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If you draw the bounded region in the x,y-plane, you'll find it to be somewhat ambiguous, but since y = 0 cuts the area between the parabola x = y ² and x = 4 perfectly in half, you can use either the top or bottom half. I'll use the top one, i.e. assume y ≥ 0.

For every x taken from the interval [0, 4], we can get a shell with height √x. The distance from x to the axis of revolution, x = 5, is 5 - x, which corresponds to the radius of the shell. The area of this shell is

2π (radius) (height) = 2π (5 - x) √x

Then the volume of the solid is the sum of infinitely many such shells made at every 0 ≤ x ≤ 4, given by the integral


\displaystyle 2\pi \int_0^4 (5-x)\sqrt x\,\mathrm dx = 2\pi \int_0^4 \left(5x^(1/2)-x^(3/2)\right)\,\mathrm dx \\\\ = 2\pi \left(\frac{10}3x^(3/2)-\frac25x^(5/2)\right)\bigg|_0^4 \\\\ = \boxed{(416\pi)/(15)}

User Danny
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