Find the volume of the solid obtained by rotating the region

Question

Find the volume of the solid obtained by rotating the region bounded by the given curves below about the line y=7. y= x^4, y=1

Answer 1

First I suggest graphing the functions. The function y=x⁴ looks almost like a parabola, while the function y=1 is a horizontal line. These two functions will define the bounded region. After graphing you'll see that the line y=1 intersects y=x⁴ at x=-1 and x=1. The region is bounded below by y=x⁴ and bounded above by y=1.

Next we can graph the axis of rotation, which is another horizontal line y=7. Since there is a gap between the axis of rotation and the closest boundary to the region, I suggest using the Washer Method. The Washer method is preferred to the Shell method because we are rotating around a horizontal axis and our functions are in terms of x, not y.

We next need to measure the radii from the axis of rotation to the furthest boundary and to the closest boundary. Let R(x) define the radius from the axis to the furthest boundary and r(x) be the radius from the axis to the closest boundary.

R(x) = 7 - x⁴ [this is the distance between y=7 and y=x⁴]

r(x) = 7-1 = 6 [this is the distance between y=7 and y=1]

Volume = π ∫ [ (7-x⁴)² - 6² ]dx

The lower value of the integral is -1, the upper value is 1.

Inside the integral: (7-x⁴)² - 6² = x⁸ - 14x⁴ +13

Integrating this function gives: 1/9*x⁹ - 14/5*x⁵ + 13x

Evaluating from -1 to 1: 1/9(1)⁹ - 14/5*(1)⁵ + 13(1) - [1/9(-1)⁹ - 14/5*(-1)⁵ + 13(-1)] = 928/45

Multiply by π, so the final answer is 928π/45