80. Volume The region enclosed by the curve y=sech(x), the x-a...

Question

80. Volume The region enclosed by the curve y=sech(x), the x-axis, and the lines x=±ln√3 is revolved about the x-axis to generate a solid. Find the volume of the solid.

Accepted Answer

The region under Y equals skin H x divided by 2 from x equals negative log 2 to x equals positive log 2 is revolved about the x axis. Compute the volume of the resulting solid. We have 4 possible answers, 3 pi divided by 23 pi divided by 42 pi divided by 3, or 4 pi divided by 3, all in cubic units. Now, let's find the volume. This is a disc method problem. So we have the volume equals the integral, may be of pi multiplied by R of X2 DX. Now, in this case, R is our radius, and this will be C and H X divided by 2. I'll move the pi outside the integral to get pi from negative log 2 to positive log 2. Of C H X divided by 22 DX. OK, now we can solve our integral. Let's do a little bit of manipulation first. We have 2 pi integral negative log 2 to log 2. Seeking H X divided by 22. With D X divided by 2. We can use this as a substitution and find the integral of sin H2. This is a known antiderivative. See and H2's antiderivative is just tangent. So we have 2 pi. Tangent H of X divided by 2 from negative log 2 to positive natural log of 2. We also know that Tangent H of X. is equivalent 2 E X minus E to the negative X. Divided by e x plus e to the negative x. So We can use this and plug in our bound to get our volume. This volume will be equal to 2 pi. Multiplied by e. See natural log, 2 divided by 2. Minus E to the negative natural log 2 divided by 2. Divided by E to the log 2, divided by 2 plus E to the negative log 2 divided by 2. Minus E to the negative log 2 divided by 2, minus E to the log 2 divided by 2. Divided by E to the negative natural log 2 divided by 2, plus E divided by. Natural log 2 divided by 2 and the exponent. Now, we can also recall. That e to the natural log x is just equals x. So we have e to the log 2 divided by 2. That would equal e to the natural log 2. To the one half, which is just square root of 2. We can now simplify a lot of our eats. We get 2 pi multiplied. By the square root of 2 minus 1 divided by the squareroot of 2. -1 divided by the square root of 2 plus the square ot of 2. Divided by the square root of 2 plus 1 divided by the square root of 2 when we combine our fractions. We cannot simplify everything here. Simplify our square it's a 2, we end up getting 2 pi. Multiplied by 2/3. Which gives us 4 pi divided by 3. In cubic units because of volume. That means the answer to our problem of 4 pi divided by 3 is answer D. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

80. Volume The region enclosed by the curve y=sech(x), the x-axis, and the lines x=±ln√3 is revolved about the x-axis to generate a solid. Find the volume of the solid.

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