**Educational Content: Polar Curves Area Calculation****Problem Statement:**Find the area of the region that lies inside both curves:1. \( r = 7 \sin(2\theta) \)2. \( r = 7 \sin(\theta) \)**Steps for Solution:**1. **Determine the Intersection Points:** - Set the equations equal to each other to find the values of \(\theta\) where the curves intersect: \[ 7 \sin(2\theta) = 7 \sin(\theta) \]2. **Solve for \(\theta\):** - Use trigonometric identities to simplify and find intersection points.3. **Calculate the Area:** - The area \(A\) inside the curves can be found using the integral: \[ A = \frac{1}{2} \int (\text{outer curve})^2 - (\text{inner curve})^2 \, d\theta \] - Determine the limits of integration from the intersection points.4. **Integrate:** - Evaluate the definite integral to find the total area.**Key Concepts:**- **Polar Coordinates:** Useful for plotting and calculating areas enclosed by curves that depend on angles.- **Trigonometric Identities:** Can simplify complex expressions and are essential for solving equations involving \(\theta\).For more detailed calculations and examples, continue exploring similar problems in polar coordinates.

Given polar curves are: First curve is r1=7sin2θ Second curve is r2=7sinθ Area of a polar curve between a and b is calculated as: 12∫abr2dθ The intersection points are obtained by equating r1=r2: 7sin2θ=7sinθ⇒2sinθcosθ=sinθ⇒sinθ=0 or cosθ=12⇒θ=0 or θ=π3 Given curves are shown in the figure below: By symmetry, area inside the curves in first quadrant is: A1=12∫0π37sin2θ2dθ+12∫π3π27sinθ2dθ=492∫0π3sin22θdθ+492∫π3π2sin2θdθ Here, use sin2θ=1-cos2θ2 in the above expression. A1=492∫0π31-cos4θ2dθ+492∫π3π21-cos2θ2dθ=494θ-14sin4θ0π3+494θ-12sin2θπ3π2 Now, apply the limits. A1=494π3-14sin4π3-0+14sin0+494π2-12sin2π2-π3+12sin2π3=494π3-14-32+494π2-0-π3+1232=494π3+38+π2-π3+34=494π2+38+34=494π2+338Now, the total area inside both the given curves is twice the area in first quadrant. That is A=2A1 A=2494π2+338=492π2+338=494π+334 Therefore, area inside the curves r=7sin2θ and r=7sinθ is: 494π+334