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The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$ x^2 + (y - 1)^2 = 1 $ ; about the y-axis

$\frac{4}{3} \pi$

Applications of Integration

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okay, we know it's specified in the problem. The radius is X and the height is two time squirt of one minus X scored. Which means we're gonna be putting into the general formula, which is two pi times the integral from 01 off the height times the radius D of acts 01 are two bounds. Therefore, we end up with V is gonna be four pi times integral from 01 ax time scored of one less ex squirt. Where did the two go? You ask? Well, we remember the height was two time squared of our mess. X squared. I simply pulled the two out on the outside. It's a constant multiplied it by the original two pi toe end up with four pi on the outside Just a simple five days. Now we know in this context we can actually use U substitution, which you learned in a previous chapter. If you is one minus X squared than taking d'you, we end up with d'you. The derivative is negative to axe de axe. So what this means is that negative 1/2 to you is ex dx other words gonna be dividing four by two which means we end up with and remember, it's actually dividing for by negative, too, because it was negative. Two ex Texas do you So negative two pi times the integral from 10 There flips now because of the nature of the fact that we've taken u substitution. We've got negative value. Do the power rule In order to integrate increased exports by one divide by the new exponents, we end up with four pi divided by three