1) What are the areas bounded by the lines and the x axis? In each case x>0 a) y = 3/xb) y=12xc) y=(1/12)x2)Find the volume V of the solid obtained by rotating the bounded area about the x axisy=4-(1/2)x, y > 0, 2< x <3
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1) a) `y = 3/x` , `x>0`
The area under the graph is given by
`int_0^infty 3/x dx = 3log(x)|_0^infty = 3[lim_(x-> infty)log(x) - lim_(x->0)log(x)] `
`= lim_(x-> infty) x`
b) `y = 12x`, `x>0`
The are under the graph is given by
`int_0^infty12x dx = 6x^2|_0^infty = lim_(x-> infty)6x^2 - 0 = lim_(x->infty) x`
c) `y = 1/(12x)` , `x>0`
The area under the graph is given by
`int_0^infty 1/12x dx = 1/24x^2|_0^infty = 1/24lim_(x->infty) x^2 - 0 ``= lim_(x->infty)x`
2) We want the volume V of the line
`y = 4 - 1/2x`, `2<x<3`
rotated about the x axis. We obtain this by integrating over discs of radius `y` from `x =` 2 to 3. The area of each disc is then `piy^2`.
`V = int_2^3 pi(4-1/2x)^2 dx = int_2^3 pi(16 - 4x + 1/4x^2)dx`
`= pi(16x - 2x^2 + 1/12x^3)_2^3 = pi(48-18+27/12 - 32+8 - 8/12)`
`= pi(6 + 19/12) = (91/12)pi`
The volume V is called a solid of revolution.
1) The areas all tend towards infinity
2) V = (91/12)pi
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