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 bounded by the x-axis and the curve y = x sin(x), 0 ≤ x ≤ π, about

a. The y-axis.

(See Exercise 57 for a graph.)

Accepted Answer

Determine the volume of the solid obtained by revolving the region bounded by the x axis and the curve Y equals X cosine x, for X between 0 and pi divided by 2, about the y axis. Now, to solve this, we'll make use of the shell method. This formula is given by V equals 2 pi integral from A to B of the shell's radius multiplied by the shell's height. We'll use the Dx for this. Now in this case, a radius. S equals the absolute value of x minus c. But since we are around the Y axis, C is just 0, so this is just X. Our height Is fx minus g x. Where FX is our topmost function. However, our only function is x cosine x. The other function would be zero, so we just have X cosine x. We can now write this integral as 2 pi from 0 to pi divided by 2. Of X cosine x multiplied by XDX. Which is just 2 pipe. Integral from 0 to pi divided by 2. X 2 cosine xDX. Now we can solve this by using integration by parts. The integration by parts is given by the integral UDV equals UV minus integral VDU. In this case, U is going to be X2, and DU will be 2 XDX, DV will be cosine XDX, and V will be sin of X. So we can plug this into our integration by parts formula. We have 2 pi multiplied by UV, which is x 2 sin x minus the integral. Sin x multiplied by 2 xDX. Now we do have bounds, but we'll plug those in later. We have 2 x sinx. This is another integration by parts. In this case, U is 2 X, DU is 2DX, DV is sin. And V is negative cosine. So, we can do this again, getting 2 pi. X 2 sin x minus parentheses. 2 X Cosine x with a negative. Minus the integral Of -2 cosine xDX. Now we can simplify. Tupa. X2 sin x. Plus 2 X cosine X. Plus the integral to cosine x d x. Now we could take the angle of cosine, which we know is just sign. So you have 2 pi multiplied by x2 sin x plus 2 x cosine x. + 2 sin of x From 0 to pi divided by 2. Now we just plug in our bounce. If we plug in all rebounds. We ended up getting Tupa Multiplied By pi 2 divided by 4. -2 0. This just gives us. Pi cubed minus 8 pi divided by 2, and because this is volume, this is in cubic units. This is the volume, and it is our answer. 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 bounded by the x-axis and the curve y = x sin(x), 0 ≤ x ≤ π, abouta. The y-axis.(See Exercise 57 for a graph.)

Watch next

Finding volume: Find the volume of the solid generated by revolving the region bounded by the x-axis and the curve y = x sin(x), 0 ≤ x ≤ π, about
a. The y-axis.
(See Exercise 57 for a graph.)