Respuesta :
The area of the region which is inside the curve r = -4 sinθ but outside r = 2 will be 4.21 square units
What is an area bounded by the curve?
When the two curves intersect then they bound the region is known as the area bounded by the curve.
The region outside the circle r = 2 and inside the circle r = - 4 sinθ.
Then the intersection point will be given as
[tex]\rm 2 = -4 \sin \theta\\\\\theta = 3.66519 ,5.7595[/tex]
Then by the integration, we have
[tex]\rightarrow \dfrac{1}{2} \int _{3.6652}^{5.7595} [(-4\sin \theta)^2 - (2)^2] dx\\\\\\\rightarrow \dfrac{1}{2} \int _{3.6652}^{5.7595} [16 \sin ^2 \theta - 4]dx\\\\\\\rightarrow \dfrac{1}{2} \int _{3.6652}^{5.7595} [ 8(1- \cos 2\theta -) - 4]dx\\\\\\\rightarrow \dfrac{1}{2} \int _{3.6652}^{5.7595} [4 - \cos 2\theta] dx\\\\\\[/tex]
[tex]\rightarrow \dfrac{1}{2} \left [ 4\theta + \dfrac{1}{2} \sin 2\theta \right ]_{3.6652}^{5.7595} \\\\\\\rightarrow \dfrac{1}{2} \left [ 4(5.7595 - 3.6652) + 0.5 (0.19969-0.12759) \right ][/tex]
[tex]\rightarrow \dfrac{1}{2} \left [ 4(2.0943) + 0.5 (0.0721) \right ]\\\\\rightarrow \dfrac{1}{2} \left [ 8.41325 \right ]\\\\\rightarrow 4.2066 \approx 4.21[/tex]
Thus, the area of the region is 3.75 square units.
More about the area bounded by the curve link is given below.
brainly.com/question/24563834
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