Calculus II Homework Question I need the work shown not sure how my teacher is getting the answer.

Question

Find the volume of the solid generated by revolving the region between the graph of y=x^2 and y=x+4 about the x-axis.

Answer 1

x² = x + 4; x² - x - 4 = 0; x = (1 ± √17)/2; a = (1 - √17)/2; b = (1 + √17)/2

Volume v = π∫_a^b[(x + 4)² - x⁴]dx = 205.523

Answer 2

The area between the two curves is an upright parabola with a slanted cut through it. When rotated around the x axis, it the volume cross-section looks like sideways angel wings and the volume itself like a lopsided butterfly yoyo. The appropriate method to use for the volume is the "washer" approach because there is a central void region and the curves are always in the same relation to each other (and are function of x)

Drawing vertical washers we want the integral of π (R_outer² - R_inner²) dx = π (y_Line² - y_para²) dx or

integral from x₁ to x₂ of π((x+4)² - x⁴)dx where x₁ and x₂ are the points of intersection:

x+4 = x² x²-x-4=0 x₁ = 1/2(1-sqrt(17)) and x₂ = 1/2(1+sqrt(17))

I have to stop here due to time constraints.