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Burka [1]
1 year ago
13

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. Only 1 try

remaining for this problem or else I get a zero. Please help if you know the answer or how to solve.

Mathematics
1 answer:
vagabundo [1.1K]1 year ago
6 0

Using the shell method, the volume is

\displaystyle 2\pi \int_0^1 (2-x) \cdot 8x^3 \, dx = 16\pi \int_0^1 (2x^3 - x^4) \, dx

Each cylindrical shell has radius 2-x (the horizontal distance from the axis of revolution to the curve y=8x^3); has height 8x^3 (the vertical distance between a point on the x-axis in 0\le x\le1 and the curve y=8x^3).

Compute the integral.

\displaystyle 16 \pi \int_0^1 (2x^3 - x^4) \, dx = 16\pi \left(\frac{x^4}2 - \frac{x^5}5\right) \bigg|_{x=0}^{x=1} \\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = 16\pi \left(\frac12 - \frac15\right) \\\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ = \frac{24}5\pi = \boxed{4.8\pi}

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