Answer to Question #183770 in Civil and Environmental Engineering for Lyndon Billones

Question

Find the volume of the solid formed by revolving the area bounded

by the parabola y2 = 9x and the line y = 3x about the line y = -2.

Answer 1

Point of intersection of y2=9x and y=3x

Since y=4.5x and y=3x

Hence 4.5x=3x which gives x=4.5 and x=3

Hence integration is to be carried out be between x=3 and x=4.5 ➝ y=-2 and y=1.5x=-3

Line is below the curve

Taking vertical strips of width dx and rotating about x-axis we the volume generated as

dV= πy²(line)-πy²(curve)=π{(1.5x)²-(x²)²}dx

or dV=π(-3x²-x⁴)dx

Hence Volume V=∫(from 3 to 4.5)π(-3x²-x⁴)

=(from 3 to 4.5)π(-3x³/3-x⁵/4.5)

=π[{(-3×4.5³/3–4.5⁵/4.5)-(-3×3³/3–4.5⁵/4.5)}]

=π{(2125/3–2125/4.5)-(0–0)}

=π(5250/13.5)=1050π/3 cubic units