Question

Find the volume of the solid that is
generated by revolving the region bounded by x=2√y
x=0 and y = 9 around the y axis ​

Answer 1

2D plot of the three equations y=x3 , Y-axis ( x=0 ) , y=3 :

If the closed area is rotated around the Y-axis, we will get something similar to :

Integrating on y:

Consider a thin rectangle of length x and height dy at some y as shown in figure.

The Volume of this rectangle if it rotated around Y-axis is (Or Volume of a cylinder with radius x and height dy ):

⟹π∗x2∗dy

Here x depends on y.

y=x3⟹x=y13

Now integrate it from y=0 to y=3.

Volume=∫30πx2dy

Volume=π∫30y23dy

Upon Simplifying :

Volume=π∗323+123+1

Volume=π∗3∗3535

Volume=π∗3835

Hope, it helps.

Step-by-step explanation:

please see the image will be helpful for you