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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2:13 minutes

Problem 5.4.29

Textbook Question

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.
ƒ(𝓍) = cos 𝓍 on [―π/2 , π/2]

Verified step by step guidance

Step 1: Recall the formula for the average value of a function ƒ(𝓍) on an interval [a, b]. The average value is given by:

\frac{1}{(b - a)} \int_{a}^{b} ƒ (𝓍) d 𝓍

Step 2: Substitute the given interval [―π/2, π/2] into the formula. Here, a = ―π/2 and b = π/2. The formula becomes:

\frac{1}{(π / 2 - - π / 2)} \int_{-}^{π} \cos (𝓍) d 𝓍

Step 3: Simplify the denominator of the fraction. The length of the interval [―π/2, π/2] is π. The formula now becomes:

\frac{1}{π} \int_{-}^{π} \cos (𝓍) d 𝓍

Step 4: Compute the integral of cos(𝓍) over the interval [―π/2, π/2]. Recall that the integral of cos(𝓍) is sin(𝓍). Apply the Fundamental Theorem of Calculus:

\frac{1}{π} [\sin (𝓍)] | - π / 2^π / 2

Step 5: Evaluate the definite integral by substituting the limits of integration (―π/2 and π/2) into sin(𝓍). Simplify the result to find the average value of the function. Finally, draw the graph of ƒ(𝓍) = cos(𝓍) on the interval [―π/2, π/2] and indicate the average value as a horizontal line on the graph.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Average Value of a Function

The average value of a function over a given interval is calculated using the formula (1/(b-a)) * ∫[a to b] f(x) dx, where [a, b] is the interval. This concept helps in understanding how the function behaves on average across the specified range, providing insight into its overall trend rather than just its individual values.

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, over a specific interval. It is denoted as ∫[a to b] f(x) dx and is essential for calculating the total value of a function between two points, which is a key step in finding the average value.

Graphing Functions

Graphing a function involves plotting its values on a coordinate system, which visually represents its behavior over an interval. This is crucial for understanding the function's characteristics, such as peaks, troughs, and the average value, allowing for a more intuitive grasp of the function's overall performance.

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Multiple Choice

Find the average value $f_{a v e}$ of the function $f (x) = 3 x^{2} + 4 x$ on the interval $[- 1, 3]$ .

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Textbook Question

Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.

(a) ∫₋₄⁴ ƒ(𝓍) d𝓍

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Textbook Question

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

ƒ(𝓍) = 𝓍³ on [―1, 1]

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Textbook Question

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

ƒ(𝓍) = 1/(𝓍² + 1) on [―1, 1]

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Textbook Question

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

ƒ(𝓍) = 𝓍ⁿ on [0,1] , for any positive integer n

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Textbook Question

Average distance on a parabola What is the average distance between the parabola y = 30𝓍 (20 ― 𝓍 ) and the 𝓍-axis on the interval [0, 20] ?

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Textbook Question

Average velocity The velocity in m/s of an object moving along a line over the time interval [0,6] is v (t) = t² + 3t. Find the average velocity of the object over this time interval.

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Textbook Question

Average height of a wave The surface of a water wave is described by y = 5 (1 + cos 𝓍) , for ― π ≤ 𝓍 ≤ π, where y = 0 corresponds to a trough of the wave (see figure). Find the average height of the wave above the trough on [ ―π , π] .

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Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.ƒ(𝓍) = cos 𝓍 on [―π/2 , π/2]

Key Concepts

Average Value of a Function

Definite Integral

Graphing Functions

Watch next

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.
ƒ(𝓍) = cos 𝓍 on [―π/2 , π/2]