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Problem 24 Hard Difficulty

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis.
(b) Use your calculator to evaluate the integral correct to five decimal places.

$ y = x $ , $ y = \frac{2x}{(1 + x^3)} $ ; about $ x = -1 $

Answer

a) 2$\pi \int_{0}^{1}(x+1)\left(\frac{2 x}{1+x^{3}}-x\right) d x$
b) 2$\pi\left(\frac{2 \pi}{3 \sqrt{3}}-\frac{5}{6}\right) \approx 2.36164$

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Video Transcript

we know that we need to first find the intersection points, which means that if acts is two acts over one plus x cubed again, we're setting. If wise acts and wise to x over one plus x cube, we're setting the why equal to the y in terms of X. So for X, we end up with acts. Times X cubed minus one is zero and factored form, which gives us two values for ax zero and one. That's good. The's There are two bounds. Now we know the radius of each cylinder is X minus negative one, which is the same thing as X plus one. We know the height is gonna be two acts over one plus X cubed minus acts. Which means now we know we can write the integral as two pi times integral from 01 of X plus one against our times H So we're literally just putting it. This is R H is two acts over one plus X cubed minus axe. So just our terms. It's just these two values plugged in times. Divac's okay, this was part A was right. The interval part B is putting into your calculator. We're gonna put it into a graphing calculator. On we get 2.36164 and you can plug in exactly what we just wrote. If you have a graphing calculator is the same thing as two pi times two pi over three spurt of three minus five Sex But you can literally just plug in this whole integral into your calculator.