Question

The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method. y = −x^2 + 12x − 35,    y = 0;    about the x-axis

Answer 1

up vote2down voteacceptedI think your limits of integration are incorrect. If you substitute y=4y=4 into y2−x2=9y2−x2=9, you find that x=±7–√x=±7. Therefore, the two curves intersect at x=±7–√x=±7. By washer method, we have:V=π∫7√−7√(4)2−(x2+9−−−−−√)2dx=2π∫7√016−(x2+9)dx=2π∫7√07−x2dx=2π[7x−13x3]7√0=2π(147–√3)=28π7–√3V=π∫−77(4)2−(x2+9)2dx=2π∫0716−(x2+9)dx=2π∫077−x2dx=2π[7x−13x3]07=2π(1473)=28π73And just for fun, let's try the shell method. Here, we have no choice but to find the volume obtained by revolving just the part of the region in the first quadrant, and doubling it.VVVVVV=2×2π∫43yy2−9−−−−−√dy=4π∫43yy2−9−−−−−√dy=4π[13(y2−9)32]43=4π[13(y2−9)32]43=4π[77–√3]=28π7–√3