Multiple Choice
Which of the following integrals correctly represents the area of the region enclosed by the curves and for ?
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9. Graphical Applications of Integrals
Area Between Curves
Multiple Choice
Find the area of the region enclosed by the inner loop of the curve .
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Verified step by step guidance1
Step 1: Recognize that the problem involves finding the area enclosed by the inner loop of the polar curve r = 2 + 4 sin(θ). The formula for the area enclosed by a polar curve is A = (1/2) ∫[r(θ)]² dθ, where the limits of integration correspond to the range of θ that defines the inner loop.
Step 2: Determine the range of θ for the inner loop. The inner loop occurs when r = 0. Solve the equation 2 + 4 sin(θ) = 0 to find the values of θ. Rearrange to get sin(θ) = -1/2, which gives θ = 7π/6 and θ = 11π/6 within the interval [0, 2π]. These are the bounds for the inner loop.
Step 3: Set up the integral for the area of the inner loop. The area is given by A = (1/2) ∫[7π/6 to 11π/6] [2 + 4 sin(θ)]² dθ. Expand the square to simplify the integrand: [2 + 4 sin(θ)]² = 4 + 16 sin(θ) + 16 sin²(θ).
Step 4: Break the integral into separate terms: A = (1/2) ∫[7π/6 to 11π/6] (4 dθ) + (1/2) ∫[7π/6 to 11π/6] (16 sin(θ) dθ) + (1/2) ∫[7π/6 to 11π/6] (16 sin²(θ) dθ). For the sin²(θ) term, use the identity sin²(θ) = (1 - cos(2θ))/2 to simplify.
Step 5: Evaluate each term of the integral. The first term involves a constant, the second term involves the integral of sin(θ), and the third term involves the integral of (1 - cos(2θ))/2. Combine the results to find the total area of the inner loop.
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