The volume of the solid generated by revolving this region about the x-axis is 33614π.
How to calculate the volume of solid generated by curves.
Given
y = 49 - x² and y = 0
find the x-values where the curves intersect:
49 - x² = 0
x² = 49
x = √49
x = +-7
set up integral to find the volume of the solid generated by revolving this region about the x-axis.
The volume V of the solid of revolution is given by the integral:
V = π∫from a to b[f(x)]²dx
where
f(x) is the height function
[f(x)]² represents the cross-sectional area at each x.
V = π∫₋₇⁷[49 - x²]²dx
= π∫₋₇⁷[2401 - 98x² + x⁴]dx
Integrate
= π[2401x - 98x³/3 + x⁵/5]₋₇⁷
= π[2401(7) - 98(7)³/3 + (7)⁵/5 -[2401(-7) - 98(-7)³/3 + (-7)⁵/5].
= π[2401(7) + 2401(7)]
= 33614π
The volume of the solid generated by revolving this region about the x-axis is 33614π.