What is the volume of the solid produced by revolving f(x)=sqrt(81-x^2) around the x-axis?

1 Answer

Volume color(blue)(V=972 pi" ")cubic units

Explanation:

Solution 1.

The given curve is located at the first and second quadrants as shown in the graph

Desmos.comDesmos.com

You will notice that the graph shows that it is a half circle with radius r=9" "units. If we revolve this about the x-axis the solid form is a sphere.

Formula for volume of the sphere

V=4/3 pi r^3

V=4/3 pi 9^3

V=4/3 pi 729

color(blue)(V=972 pi" ")cubic units

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Solution 2.

To solve for the volume by the Calculus , we make use of the Disk Method

dV=pi r^2 dh

dV=pi y^2 dx

V=int_(-9)^9 pi*y^2 dx=pi*int_(-9)^9 sqrt((81-x^2))^2 dx

V=pi*int_(-9)^9 (81-x^2) dx

V=pi*[81x-x^3/3]_(-9)^9

V=pi*[81*9-9^3/3-(81(-9)-(-9)^3/3)]

V=pi*[729-729/3-(-729+729/3)]

V=pi*[729-243+729-243]

color(blue)(V=972 pi" ")cubic units.

God bless....I hope the explanation is useful.