Volume: Find the volume of the solid generated by revolving th...

Question

Volume: Find the volume of the solid generated by revolving the region in Exercise 45 about the x-axis.

Accepted Answer

Compute the volume of the solid obtained by revolving the region bounded by a Y equals 2 cosine x above and below by Y equals 1/2 sinx over the interval of x between pi divided by 4 to pi divided by 4, about the x axis. Now we're rotating around the X axis, which means we'll use the washer method based on what we have. This is V equals. High integral from A to B. Of our outer radius squad minus our inner radius squared. DX Now in this case our outer radius. Is given by 2 cosine x and our inner radius. Is given by 1/2 seeking x. Let's go ahead and set up our integral. We have V equals the integral from gpi divided by 4 to pi divided by 4. Of 2 cosine x2 minus 1/2 sequin X2 DX. Now we can go ahead and simplify all of our terms. We also have a pie in the very fronts. You have pi integral. Of the same bounce. Up 4, cosine squared X. Midas 1/4 22 x d x. Now, let's go ahead and simplify this even further by separating our integral into two different integrals. We get pi multiplied by the integral, pi divided by 4 to pi divided by 4. Of Cosine squared XDX. I'll go ahead and bring out that 4. minus 1/4 integral. pi divided by 4 to pi divided by 4 of sequence squared XDX. Now we can integrate each term separately. Let's first look at cosine squared x. We have the integral from gpi divided by 4 to pi divided by 4, cosine squared x dx. Now we can use an identity here. We know that cosine 2 x is equal to 2 cosine square x minus 1. Which means cosine squared x if we rewrite this can be written as 1/2. Cosine 2 x + 1. So, we can write this then as 1/2 integral pi divided by 4 to pi divided by 4. Of cosine 2 x plus 1. Now we can solve this integral. We also know that cosine 2 x plus 1 is an even function. So therefore, we can manipulate our bounce as 1/2 multiplied by 2 integral from 0 to pi divided by 4, cosine to x + 1 DX. Now we can solve this integral. We get The integral 0 to pi divided by 4. Cosine to x + 1. D X This integral then I 1/2 sin to x. Plus x from 0 to pi divided by 4. From here we can plug in our bounce. We end up getting 1/2 Sine of 2 multiplied by pi divided by 4 plus pi divided by 4. -1/2 sin of 2 multiplied by 0 + 0. This gives us 1/2 plus pi divided by 4. Now let's look at our other term. We have the integral. Of sequent squared x d x Now, very similar to the previous function, this is an even function. So we can manipulate these bounds to be from 0 to pi divided by 4. This will have a 2 in the front, and we can take this integral, which is just tangent, giving us 2 tangent of x from 0 to pi divided by 4. This gives us a 2 tangent of pi divided by 4. Minus tangent of 0. Which is just a value of 2. Now, we can write our final integral. Our volume It's pi multiplied. By 4 Multiplied by 1/22 plus pi divided by 4. And then minus 1/4 multiplied by 2. We can then simplify all of this. We end up kidding. Pi multiplied by 3 halves plus pi. Which simplifies to 3 pi plus 2 pi 2 divided by 2. This will be in cubic units because it is a volume. This is the answer to our problem. I hope to help you solve it. Thank you for watching. Goodbye.

Volume: Find the volume of the solid generated by revolving the region in Exercise 45 about the x-axis.

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