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

Given the curves $y = x^{2}$ and $y = 4$ , find the area of the region bounded by these curves between $x = - 2$ and $x = 2$ .

16 - \frac{32}{3}

8

4 - \frac{8}{3}

16

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Verified step by step guidance

Step 1: Understand the problem. You are tasked with finding the area of the region bounded by the curves y = x^2 and y = 4 between x = -2 and x = 2. This involves calculating the definite integral of the difference between the upper curve (y = 4) and the lower curve (y = x^2) over the given interval.

Step 2: Set up the integral. The area between two curves is given by the formula: $ \text{Area} = \int_{a}^{b} \left[ f(x) - g(x) \right] dx $, where $ f(x) $ is the upper curve and $ g(x) $ is the lower curve. Here, $ f(x) = 4 $ and $ g(x) = x^2 $, and the interval is $ x \in [-2, 2] $.

Step 3: Write the integral expression. Substitute the functions and interval into the formula: $ \text{Area} = \int_{-2}^{2} \left[ 4 - x^2 \right] dx $. This represents the total area between the curves.

Step 4: Break down the integral. Split the integral into two parts for easier computation: $ \int_{-2}^{2} 4 dx $ and $ \int_{-2}^{2} -x^2 dx $. These can be computed separately and then combined.

Step 5: Apply symmetry. Notice that the function $ -x^2 $ is symmetric about the y-axis, meaning $ \int_{-2}^{2} -x^2 dx = 2 \int_{0}^{2} -x^2 dx $. Similarly, $ \int_{-2}^{2} 4 dx = 2 \int_{0}^{2} 4 dx $. Use this property to simplify the calculations.