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I'm working through a problem about finding the volume of a solid of revolution using the washer method, and I've hit a wall with the integration step. I set up the integral correctly for the region bounded by y = x² and y = 2x, rotated around the x-axis, but when I square the outer and inner radii and subtract, I get a polynomial I'm struggling to integrate from 0 to 2. The antiderivative for the term -x⁴ is straightforward, but I'm second-guessing my expansion for the squared terms before combining them.

You're on the right track. Outer radius is 2x and inner radius is x^2, so the washer integrand is (2x)^2 minus (x^2)^2.

Squaring those radii gives 4x^2 and x^4, so the integrand is 4x^2 - x^4. Then you integrate from 0 to 2: ∫ 4x^2 dx = 32/3 and ∫ x^4 dx = 32/5, difference is 64/15, and the volume is pi times that, i.e. 64π/15.

I’ve tripped over that same setup before—the math looks tiny, but if you freeze and track the minus sign, it all clicks.

Do you want me to walk through each antiderivative step letter by letter?

If the algebra still gnaws at you, try sketching the two curves and visually checking the radii; it often makes the squared terms feel less abstract.

One more thing I learned the hard way: the curves meet at x=0 and x=2 here, so those are your bounds; flipping them changes everything.

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