1.) Find the volume of the solid obtained by revolving the region bounded by the
ID: 2849318 • Letter: 1
Question
1.) Find the volume of the solid obtained by revolving the region bounded by the graphs of the functions about the x-axis.
Hint: You will need to evaluate two integrals. (Assume x > 0.)
y = 1 / x , y = x , y = 7x
2.) By computing the volume of the solid obtained by revolving the region under the semicircle
y = (r^2 - x^2)^.5 from x = ?r to x = r about the x-axis, show that the volume of a sphere of radius r is 4/3(pi)r^3 cubic units. (Do this by setting up the integral.)
3.) Suppose the volume of air inhaled by a person during respiration is given by
liters at a time t greater than or equal to 0 (in seconds)
V(t) = 4 ( 1 cos COS. 2 TrExplanation / Answer
y = 1 / x , y = x , y = 7x
point of intersection of y = 1 / x , y = x ==> x = 1/x
==> x2 = 1 ==> x = 1
point of intersection of y = 1 / x , y = 7x ==> 7x = 1/x
==> 7x2 = 1 ==> x = (1/7)
volume v = integral [0 to (1/7)] [ (7x)2 - x2] dx + integral [(1/7) to 1] [ (1/x)2 - x2] dx
==> v = integral [0 to (1/7)] [ 48x2] dx + integral [(1/7) to 1] [ x-2 - x2] dx
==> v = [0 to (1/7)] [ 48x3/3] + [(1/7) to 1] [ x-1/-1 - x3/3]
==> v = [0 to (1/7)] [ 16x3] + [(1/7) to 1] [ -1/x - x3/3]
==> v = 16[((1/7))3 - 0] + [ -1/1 - 13/3 - (-1/(1/7) - ((1/7))3/3)]
==> v = (16/7)(1/7) + [ -1 - 1/3 + 1/(1/7) + (1/7)(1/7)/3)]
==> v = 6.8937 cubic units
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