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 $r = 6 + 12 \sin (θ)$ .

12 π

18 π

54 π

36 π

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Verified step by step guidance

Step 1: Understand the problem. The curve is given in polar form as r = 6 + 12 sin(θ). To find the area enclosed by the inner loop, we need to identify the range of θ values where the inner loop occurs. This happens when r becomes zero, so solve 6 + 12 sin(θ) = 0 for θ.

Step 2: Solve 6 + 12 sin(θ) = 0. Rearrange the equation to find sin(θ) = -1/2. Use the unit circle to determine the values of θ where sin(θ) = -1/2. These values will give the bounds for the integral.

Step 3: Set up the area formula for a polar curve. The area enclosed by a polar curve is given by A = (1/2) ∫[θ₁, θ₂] r² dθ, where θ₁ and θ₂ are the bounds of θ determined in Step 2. Substitute r = 6 + 12 sin(θ) into the formula.

Step 4: Expand r² = (6 + 12 sin(θ))². Simplify this expression to prepare for integration. You will get terms involving constants, sin(θ), and sin²(θ). Recall that sin²(θ) can be rewritten using the identity sin²(θ) = (1 - cos(2θ))/2.

Step 5: Integrate the expanded expression for r² over the bounds of θ found in Step 2. Break the integral into manageable parts, compute each term, and sum them up. Multiply the result by 1/2 to find the area of the inner loop.