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Find the volume obtained by rotating the region bounded by the curves about the given axis.

$ y = \sin x $ , $ y = 0 $ , $ \frac{\pi}{2} \le x \le \pi $ ; about the x-axis

$$V=\frac{\pi^{2}}{4}$$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

Campbell University

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this problem is from chapter seven section to problem number sixty one in the book Calculus Early Transcendental. Lt's a condition by James Door here we like to find the volume obtained by rotating the region bounded by the curves. Why equal sign X y equals zero xon between pie over to and pie. And we like to rotate this about the x axis over here on the right, we have a graph of y equals sine X for X between pyro too and pie. This is in blue. We also have in red y equals zero and then we have the bounds. X equals pi over to this is in green. This is the line vertical line X equals power too. And then our upper bound for exes over here at pie so we could see that the region bounded by all of these curves Is the area inside here? This object will seek a fin So it's commented, called Eurasian buy Some letter was calling our and moreover, rotating this are this region about the X axis. So some notation that used might see here is something like that to let us know that we're rotating this thing around the X axis so we could go out and actually draw what the solid would look like after the revolution after the rotation. So after we do so we get an object that looks like this, and then we can go ahead and make it look like a solid. So this thing with Wade you right here this our disk This is a cross section and we see that it's a washer, so let eh be the washing area. And we know that our formula for the volume the is that integral of the area of the washer. So it will be the inner world, eh? So we need to be more specific here and our problem. The is a washer. So the area will be pi r square. That's our into grand. And this problem we obtained the volume when this washer moves in the extraction. So we should be integrating with respect to X. And in doing so, we should use the bounds pirates who and pie for limits of integration. Also, we need to find our which is the radius of the washer. So are the washer radius. So let's go to our picture here. Let this be our starting point. So we see that this is the center of the washer, the center of the disk. And if we move straight up, that's a radius right there. And this is just some why value. So our equals Why we don't want to use why in the general, Because we're in a gated with respect eggs. So we use the fact that here this why, Val, you were on the blue curve. So why is given by signing books? It would be better to use this because now we're in terms of X. So are integral becomes less claw that pie. It's a constant power to the pie. And then we have signed squared of X. To evaluate this inaugural it'LL be best to use an identity a triggered a metric identity for sign square. We can write this as Science Square is one minus coastline of two x, all divided by two. And since two was just the constant, we could also pull that to a outside of the integral. And we can evaluate each of these in a girl's and a girl. One is X, and the integral of coastline of two weeks is signed two eggs over too. Where am I help you to use a U substitution here. If you can't see this for this integral, you could try and use up. U equals two X. Can I look and let's not forget our end points piracy Tobi. So it's good and plug these in for X if we plug in pie first. So let's do that. So we have pie minus sign of two pi over too. Then we plug in pi over two for X minus. Sign a pie over to you. Separate this from our scratch work on the side. So sign of too high from the unit circle. We know that zero sign of pies also zero. So we're left over with hi over too. And then pie minus pi over too. It's also a pie over too. And so we got our final answer for the volume. Pi squared over four and there's our answer

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