00:01
In the given question we have to find the area of surface obtained by rotating the curve.
00:20
X is equal to 3 times e to the power 2 times by from y is equal to 0 to y to y is equal to 1 about y axis.
00:34
So to simplify this question we have the given curve is x is equal to 3 times e to the power 2 times y if we differentiate this curve we get d x over d y is equal to 3 times the derivative of e to the power y will be e to the power 2 times y and derivative of 2 times y will be 2 that gives us 6 times e to the power 2 times y so we can give the area is equal to under root of 1 plus d x over dy whole square with respect to y.
01:22
So we get the value of d of d of s is equal to under root of 1 plus 6 times e to power 2 times y whole square with respect to dy which is equals to under root of 1 plus 36 times e to the 4 times y with respect to 1 .3 times y with respect to 1.
01:41
Now we have to find the surface area.
01:46
So the surface area is given by s is equal to integral from 0 to 1 2 times by times x with respect to dx.
01:58
So let us substitute the value of ds so integration from 0 to 1 2 times 5 and we have x is 3 times e to the power 2 times y and we calculate dss as under root of 1 plus 36 times e to the power 5.
02:14
4 times y with respect to dy which gives us s is equal to 6 times 5 integral from 0 to 1 e to the power 2 times y under root of 1 plus 36 times e to the power 4 times y with respect to dy so first we solve the integral that is integral let's find the value of integral e to the power 2 times y under the root of 1 plus 36 times e to the power 4 times y with respect to d .y.
02:52
Now here we can do one adjustment or we can rewrite it as integration of e to the power 4y over e to the power 2y times under root of 1 plus 36 times e to power 4 times y with respect to dy.
03:10
Now to simplify this we will consider that 1 plus 36 times e to the power 4 times y is equal to t square.
03:21
So if we simplify it, we get e to the power 2 times y whole square is equal to t square minus 1 over 36.
03:34
So we get e to the power 2 times y will be equal to under root of 2 x...