y=5x^2, 3x+y=8, x=0 rotated around the line x=-4
in Calculus Answers by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

The picture shows the parabola and line and the axis of rotation.

We can split the volume into parts, starting with the tip of the parabola. This volume can be calculated by considering infinitesimally thin horizontal discs of radius x+4 and thickness dy, volume π(x+4)^2dy, between limits [-1 to 1]. Since y=5x^2, dy=10xdx. Thus we arrive at V=10π∫(x(x+4)^2dx) = 10π∫((x^3+8x^2+16x)dx) = 10π[x^4/4+8x^3/3+8x^2] for -1≤x≤1 = 10π(1/4-1/4+8/3-(-8/3)+8-8)=160π/3.

For the remaining part of the volume for -1.6≤x≤-1 we subtract the inner volume created by the parabola from the outer volume created by the straight line (frustum). Vi (inner volume)=10π[x^4/4+8x^3/3+8x^2] for -1.6≤x≤-1=10π(11.1957-5.5833)=56.124π=176.3187. Vo (outer volume)=π∫(x+4)^2dy. Since y=8-3x, dy=-3dx so Vo=-3π∫((x+4)^2dx)=-3π∫((x^2+8x+16)dx) for -1.6≤x≤1. Vo=-3π[x^3/3+4x^2+16x] for -1.6≤x≤1=111.176π. Vo-Vi=(111.176-56.124)π = 55.052π = 172.95 approx.

(The volume of a frustum (truncated cone) is (1/3)π(R^2H-r^2h) where R is the base radius and r the top radius, H is the height of the complete cone and h the height of its tip (which is removed). H=20-5=15 because the line meets x=-4 at y=20 (-12+y=8) and h=20-12.8=7.2. R=5 and r=-1.6-(-4)=2.4. So the volume of the frustum is (1/3)π(25*15-5.76*7.2)=13897π/125=111.176π=349.2697 approx. This is also the calculated value of Vo using calculus.)

Combining this with the volume of the tip we get (55.052+160/3)π=108.3853π=340.50 cu units approx.

by Top Rated User (957k points)
reshown by

Related questions

1 answer
0 answers
0 answers
asked Apr 24, 2013 in Calculus Answers by anonymous | 283 views
1 answer
asked Mar 10, 2013 in Calculus Answers by anonymous | 352 views
Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!
86,943 questions
95,225 answers
24,293 users