Integration: Volume by rotation about the line x =1

y= x^(1/4)
y=x

I used the washer method and kept getting a negative value. the answer is 17pi/45

Please help me.

Volume = Integral from a to b (2*pi*r*h dx) as a general formula.

a = starting point of rotation = 0 (Mostly the origin)
b = ending point of rotation = 1 (Where it revolves around)
r = radius of rotation = 1 - x (The distance from the origin to the rotation line then minus x)
h = height of the area = x^(1/4) - x (Easy way to remember: Height of the area for dx is always the function on top minus the function on the bottom)

Volume = Integral from 0 to 1 (2*pi*(1 - x)*(x^(1/4) - x) dx)

Since 2*pi is a constant, it can be taken out of the integral.

Volume = 2 * pi * Integral from 0 to 1 ((1 - x) * (x^(1/4) - x) dx)

Simplify the integral: Volume = 2 * pi * Integral from 0 to 1 ((x^(1/4) - x - x^(5/4) + x^2) dx)

Integrate the integral: Volume = 2 * pi * ((4/5)x^(5/4) - (1/2)x^2 - (4/9)x^(9/4) + (1/3)x^3) from 0 to 1

Since every function has x in it, 0 times anything is 0 so the second part is 0.

Volume = 2 * pi * ((4/5)*(1)^(5/4) - (1/2)*(1)^2 - (4/9)*(1)^(9/4) + (1/3)*(1)^(3))

Simplify: Volume = 2 * pi * ((4/5) - (1/2) - (4/9) + (1/3))

Get the same denominator: Volume = 2 * pi * ((216 - 135 - 120 + 90) / 270)

Simplify: Volume = 2 * pi * (51/270) = 51 * pi / 135 = 17 * pi / 45