The volume will be 107.23.
Volume
It is a 3-dimension space enclosed by the boundary or occupied by the object.
Given
y = −x² + 10x − 21,
To find
The volume v of the resulting solid by any method.
How to find the volume?
The region is bounded by y = −x² + 10x − 21 and y = 0.
for the limit y = −x² + 10x − 21, in this y becomes 0. then
−x² + 10x − 21 = 0
( x - 3 ) ( x - 7) = 0
So limit will be 3 to 7.
Then the volume will be.
[tex]\rm Volume = \pi \int\limits^7_3 {y^{2} } \, dx \\\rm Volume = \pi \int\limits^7_3 { (-x^{2} +10x-21)^{2} } \, dx \\\rm Volume = \pi \int\limits^7_3 {(x^{4} -20x^{3}+121x^{2} -420x+441) } \, dx \\\rm Volume = \pi [ \dfrac{x^{5} }{5} - 5x^{4}+ \dfrac{121}{3}x^{3} - 210x^{2} +441x]_3^7[/tex]
[tex]\rm Volume = \pi [ \dfrac{(7^{5}-3^{5}) }{5} - 5(7^{4}-3^{4}) + \dfrac{121}{3}(7^{3} - 3^{3} ) - 210(7^{2}-3^{2} ) +441(7-3)]\\\\\rm Volume = \pi [ 3312.8 -11600 +12745.33-8400+1764] \\\\\rm Volume = 107.23[/tex]
Thus the volume will be 107.23.
More about the volume link is given below.
https://brainly.com/question/1578538
