Volume of solids roatated around an axis

Question

Find the volume of the solid obtained by rotating the region in the first quadrant bounded by y=x^3, y=1, and the y-axis about the line y=-4 .

Answer 1

Volume of solids rotated around an axis

Find the volume of the solid obtained by rotating the region in the first quadrant bounded by y=x^3, y=1, and the y-axis about the line y=-4 .

Hello Josh,

To solve, I highly recommend you graph the functions and apply the given parameters. Rotate the region about the line y = -4 to arrive at the solid and determine how best to approach the integration. Upon, graphing, rotating, and inspection, i decided that the disc method using partitions of width Δx would suffice for the integral.

Outer Radius

It appears that the outer radius of the solid generated (or height of the rectangles in the shaded region) is a constant of 5. That's because it is the distance from the outer edge of the solid, to the axis of rotation. Check your graph.

Inner Radius

The inner radius of the solid generated is dynamic and will change as a function of x over the interval from 0 to 1. The inner radius can be calculated by taking the y-values at each point on the curve and adding a 4 to them, since that is the distacne between the axis of rotation and the inner radius of the solid. Check graph.

Volume Integral

Using the disc method:

V = π ∫(R² - r²)Δx

V = π ∫_o¹[ 5² - (x³ + 4)² ]Δx

V = π ∫_o¹[ 25 - (x³ + 4)(x³ + 4) ]Δx

V = π ∫_o¹[ 25 - (x⁶ + 8x³ + 16) ]Δx

V = π ∫_o¹[ 25 - x⁶ - 8x³ - 16) ]Δx

V = π [ 25x - 1/7x⁷ - 2x⁴ - 16x ]_o¹

V = π [ 9x - 1/7x⁷ - 2x⁴ ]_o¹

V = π [ 9(1) - 1/7(1)⁷ - 2(1)⁴ ] - [ 0 ]

V = π [ 9 - 1/7 - 2 ]

V = π [ 63/7 - 1/7 - 14/7 ]

V = π [ 48/7 ]

V = 48/7 π

Lets Go Brandon!