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

Average Value of a Function

Calculus

8. Definite Integrals

Average Value of a Function

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Multiple Choice

Let $f (x, y) = x^{2} + y^{2}$ . Find the average value of $f$ over the rectangle with vertices $(0, 0)$ , $(2, 0)$ , $(2, 1)$ , and $(0, 1)$ .

\frac{5}{2}

3

\frac{7}{3}

2

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Verified step by step guidance

Step 1: Recall the formula for the average value of a function f(x, y) over a region R. It is given by:

f_{avg} = \frac{1}{\text{Area of } R} \int \int_R f(x, y) \, dA

. Here, the region R is the rectangle with vertices (0, 0), (2, 0), (2, 1), and (0, 1).

Step 2: Calculate the area of the rectangle R. The width of the rectangle is 2 (from x = 0 to x = 2), and the height is 1 (from y = 0 to y = 1). Thus, the area of R is

2 \times 1 = 2

Step 3: Set up the double integral to compute the total value of f(x, y) over the rectangle. The function is

f(x, y) = x^2 + y^2

, and the limits of integration are 0 to 2 for x and 0 to 1 for y. The integral is:

\int_0^1 \int_0^2 (x^2 + y^2) \, dx \, dy

Step 4: Evaluate the inner integral with respect to x first. For

\int_0^2 (x^2 + y^2) \, dx

, treat y^2 as a constant. The result will be a function of y.

Step 5: Evaluate the outer integral with respect to y. Once the inner integral is computed, integrate the resulting expression over the interval y = 0 to y = 1. Finally, divide the result by the area of the rectangle (2) to find the average value of f(x, y).