Whats the area of a region in the first quadrant between the graph of #y= xsqrt(4-x^2)# and the x axis?

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Answer 1 · 2016-12-01T14:25:31

The area is #A=8/3#.

The domain of the function #y(x)# graph{xsqrt(4-x^2) [-10, 10, -5, 5]} is:

#(4-x^2) > 0#

or #-2 <=x <= 2#

so the area between the graph and the x axis in the first quadrant is:

#A=int_0^2 xsqrt(4-x^2)dx#

Substitute #x=2sint#

#A=int_0^(pi/2) 2sintsqrt(4-4sin^2t)*2costdt#

#A=8 int_0^(pi/2) sintcostsqrt(cos^2t)*dt#

In the interval #[0,pi/2]# #cost >0#, so #sqrt(cos^2t) = cost#

#A=8 int_0^(pi/2) sintcos^2tdt=-8int_0^(pi/2) (cost )^2d(cost)#

#A=-8/3 (cos^3 t) |_(t=0)^(t=pi/2)=8/3#