# calculus

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 = −x2 + 7x − 12, y = 0; about the x-axis

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1. y = -(x-3)(x-4)

So, using discs of thickness dx,

v = ∫[3,4] πr^2 dx
where r=y
v = ∫[3,4] π(x^2-7x+12)^2 dx = π/30

Using shells of thickness dy, and taking advantage of the symmetry,

v = 2∫[0,1/4] 2πrh dy
where r=y and h=√(1/4 - y)
v = 2∫[0,1/4] 2πy√(1/4 - y) dy = π/30

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2. A variable force of
7x−2
pounds moves an object along a straight line when it is x feet from the origin. Calculate the work done in moving the object from
x = 1
ft to
x = 19

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