Let R be the region bounded by the following curves. Find the ...

Question

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=sin xon [0,π] and y=0 ; about the x-axis (Hint: Recall that sin^2 x=1 − cos2x / 2.

Accepted Answer

Let T be the region bounded by Y equals 2 of X on the interval from 0 to pi and Y equals 0. Find the volume of the solid generated when T is revolved about the X axis. We have 4 possible answers, being piquad, 4 squad, 2 squared, or 2 pi all in cubic units. Now to solve this, we'll make use of the disc method, which tells us our volume. Will be given by a pi multiplied by the integral. May to be of our function F of X squared DX. Now our bounds are given by 0 to pi. And our function is to sin x. Let's go ahead and simplify this. We can square everything to get 4, squared X, DX at the end. Now, we do know an identity, which tells us that sin squared X is equals to 1 minus cosine 2 x divided by 2. So we can substitute that identity here. We get 4 multiplied by 1 minus cosine 2 X divided by 2. DX Which is pi multiplied by the integral from 0 to pi, of 2 multiplied by 1 minus cosine, 2 XDX. We can factor out our 2 to get 2 i. Multiplied by the integral from 0 to pi of 1 minus cosine 2 XDX. And now here we can take our antiderivative. This will give us X. Minus 2 x divided by 2. From 0 to pay. From here we can just plug in our bounds. We get 2 pie multiplied. By i minus sign of 2 pi divided by 2. 0 minus the sign of 0 divided by 2. This simplifies, just give us 2 pi multiplied by pi, or 2 pi squared. This is in cubic units. This is the answer to our problem. And if we look at our possible answers, this is answer C. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.y=sin xon [0,π] and y=0 ; about the x-axis (Hint: Recall that sin^2 x=1 − cos2x / 2.

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Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.
y=sin xon [0,π] and y=0 ; about the x-axis (Hint: Recall that sin^2 x=1 − cos2x / 2.