00:01
Hi, in this question we need to compute the area of the part of the surface y is equal to x times z that lies within the cylinder x squared plus y square is equals to 1.
00:13
As we know in this case, area can be calculated as equals to double integral under root of 1 plus curly y by curly x square plus curly y by curly z square.
00:29
So to apply this formula, first we find the value for curly y by curly x that would be equals to z and the value for curly y by curly z that would be equals to x.
00:42
Now using the polar coordinates, as we know, 0 is less than equal to theta would be less than equal to 2 pi.
00:53
R will vary from 0 to 1 and x can be substituted as r cost theta and z can be substituted as r cost theta and z can be substituted.
01:02
As are sine theta.
01:05
Now substituting all these values we get our integral that would be equals to a that is integration from 0 to 2 pi integration from 0 to 1 under root of 1 plus r square cos square theta plus r square sine square theta times r d r d theta.
01:32
Solving this further we get this to be equals to integration from 0 to 2 pi integration from 0 to 1 r times under root of 1 plus r square because sine square theta plus cause square theta will become 1 times d r d theta now to solve this integral we will use the substitution for 1 plus r square to be equal to t that means 2r dr will become d t...