Question #189079

Find the area of the curve r^2 = 3-4cos^2∅, symmetrical to origin and Ox.


1
Expert's answer
2021-05-07T12:32:16-0400

Given, the curve r2 = 3-4cos2∅, symmetrical to origin.

Area of region above the origin=

=12π65π6r2d=12π65π634cos2d=12π65π6341cos22d=12π65π61+cos2d=12[+sin22]π65π6=12[5π634π634]=12[2π332]=0.614square units=\frac{1}{2}\int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} r^2 d\varnothing\newline =\frac{1}{2}\int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} 3-4cos^2\varnothing d\varnothing\newline =\frac{1}{2}\int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} 3-4\frac{1-cos2\varnothing}{2} d\varnothing\newline =\frac{1}{2}\int_{\frac{\pi}{6}}^{\frac{5\pi}{6}} 1+cos2\varnothing d\varnothing\newline =\frac{1}{2}[ \varnothing + \frac{sin2\varnothing}{2} ]_{\frac{\pi}{6}}^{\frac{5\pi}{6}}\newline =\frac{1}{2}[ \frac{5\pi}{6} -\frac{\sqrt3}{4} -{\frac{\pi}{6}}-\frac{\sqrt3}{4}]\newline =\frac{1}{2}[ \frac{2\pi}{3} -\frac{\sqrt3}{2}]\newline =0.614 \text{square units}

Total area=2×0.614=1.228 square units.



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