Let R be the shaded region bounded by the graphs of y=sqrt of x and y=e^-3x and the

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Betsy

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verttival line x=1, as sho? a) find the area of R
b) find the volume of the solid generated when R is revolved about the horizontal line y=1
c) the region R is the base of a solid. for this solid, each cross section perpendicular to the x-axis is a rectangle whose height is 5 times the length of its base in region R. Find the volume of this solid.
 
a)
Draw a graph of the functions. In the range 0 to 0.2387 , e^-3x > sqrt(x) and in the region 0.2387 to 1, sqrt(x) > e^(-3x)

The two curves intersect at x=0.23873413
Area = ? [e^(-3x) - x^(1/2) ] dx , limits [0, 0.23873413] + ? [x^(1/2) - e^(-3x)] dx, limit [ 0.23873413,1]

= -(1/3) e^-3x - (2/3) x^(3/2) limits [0, 0.23873413] = -0.240632 - (-0.33333) = 0.092698
plus
(2/3) x^(3/2) + (1/3) e^(-3x) limits [ 0.23873413, 1] = 0.6832624 - (0.2406324) =0.44263
Area =0.092698 + 0.44263
=0.535328

b)
Volume = ? ? (1-sqrt(x))^2 - (1-e^(-3x))^2 dx , limit 0 to 1
You can complete the integration
 
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