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

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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

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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.