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 Calculus Calculus Math Forum

 May 1st, 2017, 07:02 PM #1 Senior Member   Joined: Apr 2008 Posts: 194 Thanks: 3 volume of revolution A region is bounded by the curve y=x^3+x+1, x=1 and the x-axis. Find the exact volume of the solid by revolving the region about the line x=1. my solution I use the cylindrical method to solve the problem. The radius is r=1-x and the height is h=x^3+x+1. Then, I integrate 2*pi*(1-x)(x^3+x+1) from the x-intercept on the negative side of the x-axis to 1. Now, I have a problem. I don't know how to find the x-intercept. The exact value of the volume is 43*pi/30. Can someone explain how to determine the x-intercept? Thanks a lot. May 1st, 2017, 07:41 PM #2 Math Team   Joined: Jul 2011 From: Texas Posts: 3,034 Thanks: 1621 something's not right ... using a calculator, $x^3+x+1 = 0$ at $x \approx -0.6823278038$ let $a = -0.6823278038$ $\displaystyle 2\pi \int_a^1 (1-x)(x^3+x+1) \, dx \approx 7.598$ $\dfrac{43\pi}{30} \approx 4.503$ edit ... note that $\displaystyle \int_0^1 (1-x)(x^3+x+1) \, dx = \dfrac{43\pi}{30}$, which would be the region in quadrant 1 between the cubic curve, the x-axis, and $x=1$. Last edited by skeeter; May 1st, 2017 at 08:18 PM. May 2nd, 2017, 02:06 PM   #3
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I got a different answer for the integral from a to 1, but it is still off from the desired answer.
Attached Images problem.png (8.1 KB, 9 views) May 2nd, 2017, 05:20 PM   #4
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Wolfram & my TI-84 say about 7.6 ...
Attached Images Wolfram_Int.jpg (20.8 KB, 1 views) TI_defInt.png (1.9 KB, 9 views) May 2nd, 2017, 09:55 PM #5 Member   Joined: Feb 2015 From: Southwest Posts: 96 Thanks: 24 Well crap, I put a zero after the decimal. Tags revolution, volume Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post turtle Calculus 1 August 6th, 2016 02:39 PM Jmun Pre-Calculus 11 September 5th, 2015 10:09 AM Monox D. I-Fly Calculus 8 February 14th, 2015 04:11 PM marcoesteves Calculus 6 January 16th, 2011 08:15 AM FishingMad Algebra 0 December 31st, 1969 04:00 PM

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