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

8. Definite Integrals

Introduction to Definite Integrals

Calculus

8. Definite Integrals

Introduction to Definite Integrals

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Multiple Choice

Given the parametric equations $x = \sin^{2} (t)$ , $y = 4 \cos (t)$ for $0 \leq t \leq π$ , what is the area enclosed by the curve and the y-axis?

0

8

2

4

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Verified step by step guidance

Step 1: Recall that the area enclosed by a parametric curve and the y-axis can be calculated using the formula: $ A = \int y \frac{dx}{dt} dt $, where $ y $ is the parametric expression for $ y $, and $ \frac{dx}{dt} $ is the derivative of $ x $ with respect to $ t $.

Step 2: Compute $ \frac{dx}{dt} $ by differentiating $ x = \sin^2(t) $ with respect to $ t $. Use the chain rule: $ \frac{dx}{dt} = 2\sin(t)\cos(t) $.

Step 3: Substitute $ y = 4\cos(t) $ and $ \frac{dx}{dt} = 2\sin(t)\cos(t) $ into the area formula: $ A = \int_{0}^{\pi} 4\cos(t) \cdot 2\sin(t)\cos(t) dt $. Simplify the integrand to $ A = \int_{0}^{\pi} 8\sin(t)\cos^2(t) dt $.

Step 4: To simplify further, use the trigonometric identity $ \cos^2(t) = \frac{1 + \cos(2t)}{2} $. Substitute this into the integrand: $ A = \int_{0}^{\pi} 8\sin(t) \cdot \frac{1 + \cos(2t)}{2} dt $. Expand the expression to $ A = \int_{0}^{\pi} 4\sin(t) dt + \int_{0}^{\pi} 4\sin(t)\cos(2t) dt $.

Step 5: Evaluate each integral separately. For $ \int_{0}^{\pi} 4\sin(t) dt $, use the standard integral of $ \sin(t) $. For $ \int_{0}^{\pi} 4\sin(t)\cos(2t) dt $, use trigonometric identities or integration techniques. Combine the results to find the total area.