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Hello everyone.
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The given problem is in the concept of area under the curves.
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In the problem we have two curves, the first one being a circle that is r equal to 3 sine theta.
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And the second one is given in the cardioid form which is r equal to 1 plus sine theta.
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To solve the problem we take the graphical representation of the given curves.
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In the blue outline we have the circle which is r equal to 3x theta and in the green outline we have the cardiolid that is r equal to 1 plus santa.
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We first need to find the intersecting points that is given in the form of a and b now we can equate two equations that is given in the form of r therefore we can write r equal to 3 sine theta that is also equal to 1 plus sine theta therefore simplified form will give us the value of 2 sine theta being 1 which gives sine theta being 1 divided by 2 from the graphical representation we can see that the intersecting points are in the first coordinate and second coordinate therefore we have to find the intersective point in the theta value of 0 to pi.
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Now for theta which obviously has to belong in the region of 0 to pi and has the value that represents sine theta being 1 divided by 2, we can find the value of 3ta in the form of pi divided by 6 and 5 pi divided by 6.
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Now we put the value of 3ta in the given equation and that will give us the r value being 3 divided by 2 for both these values.
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Therefore, the two intersecting points, these are 3 divided by 2, 5 divided by 6, and the second one be 3 divided by 2, 5 5 divided by 6.
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Now we have to find the area that is inside the circle and outside the cartilage.
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So the shaded area that is under the yellow color, it will be the required area here.
01:57
Now the given curves are the polar curves and to find the area we can use the formula that will be equal to a is 1 divided by 2...