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Problem 10 Medium Difficulty

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis.

$ y = \sqrt{x} $ , $ x = 0 $ , $ y = 2 $

Answer

$V=8 \pi$

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Video Transcript

through this question, it could be helpful to draw a diagram. As you can see, the shaded area would be like right over here. R D of Axe would be like this. And we know axis of rotation is like this, which means now we have the inner radiance. Escort of Ex Outer Radius is too Now, remember, the volume for the washer is gonna be pi times out already a squared times DX minus pi times inner radius squared times D x. Okay, now that we've cleared this up, let's simplify for Pied de Jax minus pi axe dx. Okay, now we're ready to put this into the integral. Volume is pie. That's our constant. That's what's on the outside bounds or 04 four minus X is our function DX. When we integrated, we know we can use the power method, which means we increased the expletive by one divide by the new exponents. X becomes Xcor divide by two now at the point where we can plug in our bounds. Pi times princey 16 minus four squared divided by two which gives us eight pi