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Find the exact area of the surface obtained by rotating the curve about the x-axis.

$ x = 1 + 2y^2 $ , $ 1 \le y \le 2 $

$\frac{\pi}{24}(65 \sqrt{65}-17 \sqrt{17})$

Applications of Integration

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Campbell University

Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

The first thing we should do is we should differentiate with respect to why So if we have four why we square this we end up with 16. Why squared? Which means we now have integral from wonder, too, to pi times. Why Times square off one plus 16 y squared Do you want now a little bit of U substitution If you was one for 16 where I squared What's under the square root? Then D'You is gonna be 32. Why di roi, which means the limits of integration. Now change. We now have power over 16 on the outside and we have 17 to 65 squirt of u d y. Okay, time to integrate. When we integrate, we know we're gonna be using the power rule, which means we increase the exponents by one, and then we divide by the new extra plug it in are two bounds have been given and we know this could be written is our solution because they've said exact area so we can leave our exact area as this