Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

9. Graphical Applications of Integrals

Area Between Curves

Calculus

9. Graphical Applications of Integrals

Area Between Curves

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Multiple Choice

Find the area of the region enclosed by the inner loop of the curve $r = 2 + 4 - \sin θ$ .

\frac{9 π}{2}

2 π

2 π - \frac{9 \sqrt{3}}{2}

4 π

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Verified step by step guidance

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.