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The given curve is rotated about the y-axis. Find the area of the resulting surface.

$ x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1 $ , $ 0 \le y \le 1 $

$$

\frac{6}{5} \pi

$$

Applications of Integration

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Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

this question asked us to use the bounds of 01 and find the area of the resulting surface. Given the fact that the curve is retained about the Y axis. What we know is we're looking from 0 to 1 is our two bounds and then we have two pi. We have the original expression and then we have this multiplied by the square root of one plus one minus y two. The 2/3 over Why the 2/3 do you? Why, We know he could write this out as to pie times the integral from 0 to 1. The reason why two pies on the outside is because estate in the textbook, because it is a constant, it doesn't need to be integrated. It could just be pulled out and then be multiplied by what's in the end. Okay, What we know we now have is U substitution if you is one minus one of the 2/3 member usually used u substitution with what's in parentheses. As you can see over here, do you is negative 2/3 wide to the negative 1/3 d Y. This now means that we have negative 6/5 spy times and then our upper minus or lower bound. This is equivalent to six over five pi the negative negative cancer.