# Similar to area between two curves when you calculate

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Note: Be very careful about finding all points of intersection between two curvesExample 4: Find all points of intersection of the curves r = cos 2θ and r = 1/2More Challenging Example:Example 5: Find the area of the region enclosed by the circle r = 1/2 and the curve r = cos 3θ in the first and fourth quadrantArc LengthIf we take our polar equations x = r cos θ and y = r sin θ, apply the derivatives via product rule with respect to θ, and plug in to the parametric definition for arc length:Example 6: Find the length of the cardioid r = 1 + sin θPractice Problems1. Find the area of the region within the cardioid r = 1 + cos θ that is above the x-axis
2. Sketch the curve r = 3(1 + cos θ) and find the area that it encloses3. Find the area enclosed by the loop of the strophoid r = 2 cos θ – sec θ 4. Find the area of the region that lies inside the curve r = 3 sin θ and outside the curve r = 2 – sin θ 5. Find the area that lies inside both curves r2= sin 2θ and r2cos 2θ (lemniscates)6. Find the area inside the larger loop and outside the smaller loop of the limaçon r = 1/2 + cos θ7. Find all points of intersection of the curves r = 1 + sin θ and r= 3 sin θ8. Find the exact length of the polar curve r = e, 0 ≤ θ ≤ 2π9. Find the area between the curves r = 2 + cos 2θ and r = sin 2θ=
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