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

Evaluate the double integral by reversing the order of integration: $\int_{0}^{2} \int_{y}^{2} (x + y) d x d y$ .

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Verified step by step guidance

Step 1: Understand the problem. The given double integral is $ \int_{0}^{2} \int_{y}^{2} (x + y) \, dx \, dy $. The goal is to reverse the order of integration, which means we need to rewrite the integral with the limits of integration swapped.

Step 2: Analyze the region of integration. The inner integral $ \int_{y}^{2} $ indicates that $ x $ ranges from $ y $ to $ 2 $, and the outer integral $ \int_{0}^{2} $ indicates that $ y $ ranges from $ 0 $ to $ 2 $. This describes a triangular region in the $ xy $-plane bounded by $ x = y $, $ x = 2 $, and $ y = 0 $.

Step 3: Reverse the order of integration. To reverse the order, determine the new limits for $ x $ and $ y $. In this region, $ x $ ranges from $ 0 $ to $ 2 $, and for a fixed $ x $, $ y $ ranges from $ 0 $ to $ x $. The new integral becomes $ \int_{0}^{2} \int_{0}^{x} (x + y) \, dy \, dx $.

Step 4: Solve the inner integral. Compute $ \int_{0}^{x} (x + y) \, dy $. Since $ x $ is treated as a constant with respect to $ y $, split the integral into two parts: $ \int_{0}^{x} x \, dy $ and $ \int_{0}^{x} y \, dy $. Evaluate each part separately.

Step 5: Solve the outer integral. After evaluating the inner integral, substitute the result into the outer integral $ \int_{0}^{2} $. Integrate with respect to $ x $ to find the final value of the double integral.