Finding volume: Find the volume of the solid generated by revo...

Question

Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes and the curve y = cos(x), 0 ≤ x ≤ π/2, about

b. The line x = π/2.

Accepted Answer

Determine the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes and the curve Y equals sin of X. We also have X is between 0 and pi divided by 2, about the line X equals pi divided by 2. We have 4 possible answers being pi + 2, pi2 + 2 pi, pi minus 2, or i2 minus 2 pi, all in cubic units. Now, to solve this, we will make use of the shell method. This tells us that the volume is equal to 2 pi multiplied by the integral from A to B. Of the shell's radius multiplied by its height. DX in this case. Now radius. is given by the absolute value of X minus C, where C is the line we're revolving around. And height is given by F X minus G of X. Our radius then is the absolute value of x minus pi divided by 2, which is just pi divided by 2. Minus X Our height is just sin x, since we don't have a second function. Our second function is just the coordinate axes, 0. We cannot write out our integral. We have the integral. 2 pi in the front From 0 to pi divided by 2. Of i divided by 2 minus x multiplied. By Side of x With DX at the ends. We can distribute our signs of both terms to give us 2 pi integral from 0 to pi divided by 2. Of pi Divided by 2. Sin x minus X, sin xDX. Now from here, we can simplify and find our integral. For our first term, we just have a constant multiplied by sin x. The integral of sine is given by negative cosine. So, we have 2 pi. More off the plane. Bi pi divided by 2 negative cosine x. From 0 to pi divided by 2. Now I'll distribute the two pi to our other integral because that is an integration by parts. 2 pipe. Integral from 0 to 5 divided by 2 x sin x dx. We'll simplify our first integral first. 2 pi multiplied by pi divided by 2. That just gives us pi squared. So you have pi squared multiplied. By cosine, which is negative cosine. From pi divided by 2 to 0. So We get negative cosine pi divided by 2. Minus negative cosine of 0. Cosine pi divided by 2 is 0 based on the unit circle, and cosine of 0 is 1. We have a double negative, so we have a positive 1 multiplying our pi 2. Now let's look at our other integral. This is integration by parts. So for integration by parts, we know. That the integral UDV equals UV minus integral VDU. In this case, we will let u equals the algebraic function of x. DU is just DX. DV then will be sin x, V then will be the antiderivative, which is negative cosine x. Now we can substitute these into our formula. We have 2 pi. Multiplied By UV, which is negative cosine, X multiplied by X. Minus integral negative cosine x. D X We have then -2 pi. Multiplied By negative x cosine x. Plus the integral of cosine x when we simplify our negatives. Now the 2 pi as in multipine. Negative X cosine X. The integral of cosine is sin, so we have plus sin x. From 0, the pi divided by 2. From here, we just plug in our values. We get -2 pi multiplied by 1 when we plug in pi divided by 2 and 0. This gives us Now you live to pay. Combining our previous answer. We get pi 2 minus 2 pi. Since this is volume, it is in cubic units. If we look at our possible answers, we determine that this answer is answer D 2 minus 2 pi. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes and the curve y = cos(x), 0 ≤ x ≤ π/2, aboutb. The line x = π/2.

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Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes and the curve y = cos(x), 0 ≤ x ≤ π/2, about
b. The line x = π/2.