Question #184930

Determine the area of the region enclosed by r=√3 cos(theta) and r=sin(theta) on (π/3, 4π/3).

Expert's answer

A=∫^{π/3}_{0}∫^{sinθ}_{0}rdrdθ+∫^{π/2}_{π/3}∫^{√3cosθ}_{0}rdrdθ

=∫^{π/3}_{0}[r^{2}/2]^{sinθ}_{0}dθ+∫^{π/2}_{π/3}[r^{2}/2]^{√3cosθ}_{0}dθ

=∫^{π/3}_{0}sin^{2}θ/2 dθ+∫^{π/2}_{π/3}3cos^{2}θ/2 dθ

=1/4∫^{π/3}_{0}(1−sin2θ)dθ+3/4∫^{π/2}_{π/3}(1+cos2θ)dθ

=1/4[θ−sin2θ/2]^{π/3}_{0}+3/4[θ+sin2θ/2]^{π/2}_{π/3}

=1/4(π/3−√3/4)+3/4(π/6−√3/4)

=5/24π−√3/4

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