4. Find the volume of the solid generated by revolving about the line x = 3 the region x = 3 the region bounded by the line, the x-axis and the graph of y = 9x - 1/12 * x ^ 4

Question

Solution only

Answer 1

x=3 and y=9x-x⁴/12 meet when y=9×3-81/12=27-27/4=81/4.

The distance between any point P(x,y) on the curve and the line forms the radius of a circle and a disc is created by the rotating point. The area of the disc is π(3-x)². If the disc has a thickness dy, its volume is π(3-x)²dy so the volume of the solid created by many stacked discs=₀∫^81/4π(3-x)²dy. Since y=9x-x⁴/12, dy can be replaced by (9-x³/3)dx, and the limits changed to 0 (low) and 3 (high) for x. 

We can now evaluate the integral:

π₀∫³(3-x)²(9-x³/3)dx=

π₀∫³(9-6x+x²)(9-x³/3)dx=

π₀∫³(81-54x+9x²-3x³+2x⁴-x⁵/3)dx=

π[81x-27x²+3x³-3x⁴/4+2x⁵/5-x⁶/18]₀³=

π(243-243+81-243/4+486/5-81/2)=1539π/20=76.95π=241.75 cubic units approx.