Note that both 5 and 4 are exponents, and their product is 5*4=20, or 4*5=20. Thus,
(3^5)^4 is equal to (3^4)^5
Review:
(a to the power b)to the power c = a to the power bc.
Thus, (3 to the power 5)^4 = 3^20; the same is true if you reverse the positions of the 5 and the 4.
The graph of the parabola is now vertically translating by 3 units up
For each <em>x</em> in the interval 0 ≤ <em>x</em> ≤ 5, the shell at that point has
• radius = 5 - <em>x</em>, which is the distance from <em>x</em> to <em>x</em> = 5
• height = <em>x</em> ² + 2
• thickness = d<em>x</em>
and hence contributes a volume of 2<em>π</em> (5 - <em>x</em>) (<em>x</em> ² + 2) d<em>x</em>.
Taking infinitely many of these shells and summing their volumes (i.e. integrating) gives the volume of the region:
