_________________________________________________________________________________
Calculus  II                                                          Quiz #5                                               March 17, 2003

Name____________________                    R.  Hammack                                               Score ______
_________________________________________________________________________________

(1) Consider the region contained between the graphs of y = cos(x)^(1/2),   x = π/4,  x = π/2 and y = 0. This region is revolved around the x-axis. Find the volume of the resulting solid.

[Graphics:HTMLFiles/quiz5_5.gif]
The cross section at x is a circle of radius (cos x)^(1/2)so the cross sectional area is A(x) = π((cos x)^(1/2))^2 = π cos x.
Using the slicing formula, the volume is
V = ∫_ (π/4)^(π/2) π cos x dx = π[sin x ] _ (π/4)^(π/2) = π(sin(π/2) - sin(π/4)) = π(1 - 2^(1/2)/2) cubic units.



(2) Consider the region contained between the graphs of x = y^2 + 1,  y = 0,  y=1,  and x = 0. This region is revolved around the x-axis. Find the volume of the resulting solid.

[Graphics:HTMLFiles/quiz5_12.gif]
Using the formula for volumes by shells,
V = ∫_0^12πy(y^2 + 1) dy = 2π∫_0^1 (y^3 + y) dy = 2π[y^4/4 + y^2/2 ] _0^1 = π[y^4/2 + y^2 ] _0^1 = (3π)/2cubic units