Posted by kchik on Tuesday, December 11, 2012 at 12:54pm.
The region A is bounded by the curve y=x^25x+6 and the line y = x + 3.
(a) Sketch the line and the curve on the same set of axes.
(b) Find the area of A.
(c) The part of A above the xaxis is rotated through 360degree about the xaxis. Find the volume of the solid generated.
(d) The surface of a swimming pool takes the shape of A where the depth of the water at any point (x,y) in A is given by (x+2). Find the volume of the water in the swimming pool.

calculus  Steve, Tuesday, December 11, 2012 at 2:34pm
the curves intersect at (1,2) and (3,0)
A = ∫[1,3] (3x)  (x^25x+6) dx
= x^3/3 + 2x^2  3x [1,3]
= 4/3
the parabola dips below the xaxis on the interval (2,3), so we want to rotate between the curves on [1,2] and under the line on [2,3]:
V = ∫[1,2] pi (3x)^2  (x^25x+6)^2 dx
+ ∫[2,3] pi (3x)^2
= 12/10 pi + 1/3 pi = 23/15 pi
for the pool, multiply each strip of area by its depth:
V = ∫[1,3] (x+2)((3x)  (x^25x+6)) dx
= x^4/4 + 2/3 x^3  5/2 x^2  6x [1,3]
= 16/3
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