Find the volume of the solid obtained by rotating

Question

Let R be the region in the first quadrant bounded by the curves y = f(x) = 2x+ 1 and y = g(x) = 2x^2 − 8x + 9

Find the volume of the solid obtained by rotating the region R about the line y = 10 using a single dx-integral.

i found that pi * 1 to 4 (-x^2+5x-4)^2 - (9-x)^2 dx

i used washer method but it is not the answer i dont know.

also i want to ask how can i determine my method according to question . washer method ,disc method ,shell method.

Answer 1

The two curves meet at x = 1 and x = 4 so the integral will be from 1 to 4.

V = ∫ A(x) dx = ∫ π (r²_outer - r²_inner) dx

= ∫ π [(10 - f(x))² - (10 - g(x))²] dx

= π ∫ [(10 - 2x² + 8x - 9)² - (10 - 2x - 1)²] dx

= π ∫ [(-2x² + 8x + 1)² - (-2x + 9)²] dx

= π ∫ [(4x⁴ - 32x³ + 60x² + 16x + 1) - (4x² - 36x + 81)] dx

= π ∫ (4x⁴ - 32x³ + 56x² + 52x + - 80) dx

= π [(4/5)x⁵ - 8x⁴ + (56/3)x³ + 26x² - 80x] (from 1 to 4)

Plug in x-values from integral to get:

V = (522/5)π = 104.4π

The disc and washer method both use representative rectangles that are perpendicular to the axis of revolution. The only difference is the disc method is for shapes that don't have a hole and the washer method is for shapes that have a hole in the center. The shell method uses representative rectangles that are parallel to the axis of revolution. The disc/washer and shell method are both acceptable to use. I would use the whichever method produces the simplest integral. If you get an integral that's very complicated, try taking the inverse and see if it simplifies the equations. In this example, getting the equations in terms of y would result in fractions and square roots so it is easier to leave the equations in terms of x and integrate with respect to x.

Answer 2

I just saw the message and the answer of Janelle. She is completely right and the answer is 522 pi / 5.