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 $\int_{0}^{2} \int_{0}^{1} (2 x + 1 + x y) d y d x$ .

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Verified step by step guidance

Step 1: Understand the problem. The given double integral is $ \int_0^2 \int_0^1 (2x + 1 + xy) \,dy\,dx $. This means we are integrating the function $ 2x + 1 + xy $ first with respect to $ y $ (inner integral) and then with respect to $ x $ (outer integral).

Step 2: Perform the inner integral. To integrate $ 2x + 1 + xy $ with respect to $ y $, treat $ x $ as a constant. The integral becomes $ \int_0^1 (2x + 1 + xy) \,dy $. Break it into separate terms: $ \int_0^1 2x \,dy + \int_0^1 1 \,dy + \int_0^1 xy \,dy $.

Step 3: Compute each term of the inner integral. For $ \int_0^1 2x \,dy $, the result is $ 2x \cdot y $ evaluated from $ y = 0 $ to $ y = 1 $. For $ \int_0^1 1 \,dy $, the result is $ y $ evaluated from $ y = 0 $ to $ y = 1 $. For $ \int_0^1 xy \,dy $, the result is $ x \cdot \frac{y^2}{2} $ evaluated from $ y = 0 $ to $ y = 1 $.

Step 4: Combine the results of the inner integral. Substitute the limits of integration $ y = 0 $ and $ y = 1 $ into each term. This gives the simplified expression for the inner integral as a function of $ x $.

Step 5: Perform the outer integral. Take the result from the inner integral and integrate it with respect to $ x $ over the interval $ [0, 2] $. Break it into separate terms if necessary, and compute each term using standard integration techniques. Substitute the limits $ x = 0 $ and $ x = 2 $ to find the final value of the double integral.