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

Question

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

Answer 1

A=∫^π/3₀∫^sinθ₀rdrdθ+∫^π/2_π/3∫^√3cosθ₀rdrdθ

=∫^π/3₀[r²/2]^sinθ₀dθ+∫^π/2_π/3[r²/2]^√3cosθ₀dθ

=∫^π/3₀sin²θ/2 dθ+∫^π/2_π/33cos²θ/2 dθ

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

=1/4[θ−sin2θ/2]^π/3₀+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