00:01
Hello, let's have a look at the question.
00:03
So the question state that write and evaluate the definite integral that represents the area of the surface generated by the revolving the curve on the indicated interval about the x -axis.
00:17
So we have y is equals to x to the power 3 divided by 18 and we have that x lies from 0 to 2.
00:27
Now here we have the function graph as here we have y and x -axis and the graph will be going like this and this is the point 2.
00:42
Now here we have to rotate it like this.
00:45
So we know that the surface area is calculated using the formula integration from 0 to 2 of 2 pi y and under root 1 plus y dash whole square.
01:03
So now here we will have the values as this will be equals to integration from 0 to 2, 2 pi multiplied with x to the power 3 divided by 18 and under root of 1 plus here we will have let us calculate first dy by dx.
01:25
So that means this will be x square by 6.
01:29
So we will put the value here.
01:31
So we have x square by 6 whole square and dx.
01:35
So now we will have integration from 0 to 2 of 2 pi x to the power 3 by 18 and under root of 1 plus x to the power 4 by 36 dx.
01:50
Now here let us use substitution and we will put 1 plus x to the power 4 by 36 is equals to t...