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vova2212 [387]
3 years ago
15

Use the shell method to find the volume of the solid generated by revolving the regions bounded by the curves and lines about th

e​ y-axis. =-x^2 , y=6 - 5 x​, x=​0, for x>=0
Mathematics
1 answer:
Nezavi [6.7K]3 years ago
5 0

The two curves intersect when

-x^2=6-5x\implies x^2-5x+6=(x-3)(x-2)=0\implies x=2,x=3

The region bounded by x=0 and the two curves "ends" with a vertex where x=2. So over the interval [0, 2], we have 6-5x\ge-x^2, so that the volume is

\displaystyle2\pi\int_0^2x(6-5x+x^2)\,\mathrm dx=\boxed{\frac{16\pi}3}

That is, each cylindrical shell has a radius of x and height (6-5x)-(-x^2)=6-5x+x^2, and their contribution to the total volume is 2\pi times the radius and height.

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Answer:

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