Can you help me find the area of a cardioid by double integration?

Question

Hey, I've got a math test on Friday and I can't figure out how to find the area of a cardioid by double integration. The instructions for the problem read: Find the indicated area by double integration in polar coordinates. And the problem is 3. The area bounded by the cardioid r=1+cos(theta) . Thanks in advance.

Answer 1

In polar coordinates, dA = r dr dθ so A = ∫ ∫ r dr dθ

For limits, r goes from 0 to (1 + cos(θ)) and θ goes from 0 to 2π

Because of symmetry, you could use π for the upper limit on θ and double the integral.

Answer 2

r = 3cos? Multiplying the two components via r r² = 3.rcos? = 3x x² + y² = 3x x² - 3x + y² = 0 ending up the sq. supplies (x - 3/2)² + y² = 9/4 Circle centre (3/2, 0) and radius 3/2 the section common to the circle and cardioid (the cardioid is a "cupid's coronary heart" formed curve) incorporates 2 components. (i) One section swept out via the radius vector r = a million + cos? as ? varies from 0 to ?/3. (ii) the 2d section is the section swept out via r = 3cos? as ? varies from ?/3 to ?/2. the section S is given via S = 2 ?a million/2(a million + cos?)²d? [0 to ?/3] + 2 ?a million/2(9cos²?)d? [ ?/3 to ?/2] employing the identity cos²? = a million/2(a million + cos2?) S = ? [a million + 2cos? + a million/2(a million + cos2?)]d? [0 to ?/3] + 9 ? a million/2(a million + cos2?)d? [?/3 to ?/2] S = (3?/2 + 2sin? + sin2? / 4) [0 to ?/3] + 9/2(? + sin2? / 2) [?/3 to ?/2] S = (?/2 + ?3 + ?3 / 8) + 9/2[?/2 - (?/3 + ?3 / 4)] S = ?/2 + 9?/4 - 3?/2 + ?3 + ?3/8 - 9?3/8 S = 5?/4 + ?3 - 8?3/8 = 5?/4 + ?3 - ?3 S = 5?/4 gadgets² observe : you likely would not comprise all of those steps your self. I surely have achieved so only to make sparkling some factors.