Answer to Question #184930 in Civil and Environmental Engineering for John Prats

Question #184930

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


1
Expert's answer
2021-05-05T07:42:39-0400

A=∫π/30sinθ0rdrdθ+∫π/2π/3√3cosθ0rdrdθ

=∫π/30[r2/2]sinθ0dθ+∫π/2π/3[r2/2]√3cosθ0

=∫π/30sin2θ/2 dθ+∫π/2π/33cos2θ/2 dθ

=1/4∫π/30(1−sin2θ)dθ+3/4∫π/2π/3(1+cos2θ)dθ

=1/4[θ−sin2θ/2]π/30+3/4[θ+sin2θ/2]π/2π/3

=1/4(π/3−√3/4)+3/4(π/6−√3/4)

=5/24π−√3/4



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