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Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$ y = e^{-x^2} $ , $ y = 0 $ , $ x = -1 $ , $ x = 1 $

(a) About the x-axis(b) About $ y = -1 $

a) $V \approx 3.75825$b) $V=13.14$

02:30

Carson M.

Calculus 2 / BC

Chapter 6

Applications of Integration

Section 2

Volumes

Deema M.

February 16, 2021

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 1/ 16 x2, x = 5, y = 0; about the y-axis

Oregon State University

Harvey Mudd College

University of Nottingham

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So for this problem, 31 for part A. We want to look at the rotation about the X axis. Um, So what we're gonna end up getting is a disk with radius E to the negative X squared. So our volume since the disk is going to be, um, the integral from negative 1 to 1 of pi r squared. But in this case, our radius is e to the negative x squared, squared. So we have our d X, then we simply integrate. So what we're gonna integrate is negative 1 to 1 pi e to the negative two x squared d x, Um and we can use a calculator or anything like that to find this and what we end up getting is that this volume is approximately 3.75 83 and that's going to be in Units Cube. Because it's a volume, then what we're finding, since it actually has to be to five decimal places, we'll give this 2 to 5. So it's our five decimal places. And now for part B, we want to do it about the y axis, so it's going to be similar, although this time it's going to be a washer instead of a disk. Um, so our volume is going to be a little bit more complicated. It'll be pi times the interval, um, high times the inner role from negative 1 to 1 of negative one plus one plus e to the negative x squared, squared DX. We can calculate this out further, and what we end up getting is approximately. The volume is 13 point 14 And you can do this in your calculators yourselves. And once again, remember that some units cute.

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