Use the washer method.


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 two curves (one a U shape, the other an inverted U) intersect at the solution to the equation x^2+2=-x^2+2x+6. This reduces to the quadratic 2x^2-2x-4=0 which becomes x^2-x-2=0. This factorises (x+1)(x-2)=0 so the solutions are -1 and 2, and y=3 and 6. The bounds are limited by the requirement 0<=x<=3. Therefore the bounds for the volume captured by the two curves are 0<=x<=2.
Consider first the volume of rotation of the inverted U, the superior curve, y=-2x^2+2x+6 for 0<=x<=2. Consider the disc width dx standing on its side with radius y. The volume of the disc is (pi)y^2dx=(pi)(-x^2+2x+6)^2dx. A stack of such discs placed on its side is the volume between the curve and the x axis. So we need to integrate (pi)(x^4-4x^3-8x^2+24x+36)dx for 0<=x<=2. That is, (pi)[x^5/5-x^4-8x^3/3+12x^2+36x] for 0<=x<=2.
Now we have to do the same for the other curve y=x^2+2. The volume of the thin disc is (pi)(x^2+2)^2dx=(pi)(x^4+4x^2+4)dx. This integrates as (pi)[x^5/5+4x^3/3+4x] for 0<=x<=2.
Combine the two volumes by subtracting the latter from the former (inferior from the superior) we get (pi)[-x^4-4x^3+12x^2+32x] for 0<=x<=2. So this evaluates to (pi)(-16-32+48+64)=64(pi)=201.06 cubic units.

by Top Rated User (1.0m points)

Related questions

Welcome to, 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!
87,125 questions
96,997 answers
24,433 users