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.
ฦ(๐) = ๐โฟ on [0,1] , for any positive integer n
Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.2.65
Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ฦ and the ๐-axis. Evaluate the following integrals.
โซโโฐ ฦ(๐) d๐
Verified step by step guidance1
Identify the integral to evaluate: \(\int_{a}^{0} f(x) \, dx\). Notice that the limits of integration go from \(a\) to \(0\), which is in the reverse order of the usual left-to-right direction on the x-axis.
Recall the property of definite integrals that reversing the limits changes the sign: \(\int_{a}^{0} f(x) \, dx = -\int_{0}^{a} f(x) \, dx\).
Look at the graph and observe the area between \(x=0\) and \(x=a\). The shaded region above the x-axis has an area of 16, so \(\int_{0}^{a} f(x) \, dx = 16\) because the function is positive there.
Use the property from step 2 to write \(\int_{a}^{0} f(x) \, dx = -16\) since the integral from 0 to a is positive 16 but the limits are reversed.
Thus, the value of the integral \(\int_{a}^{0} f(x) \, dx\) corresponds to the negative of the area between \(0\) and \(a\) under the curve \(f(x)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral as Net Area
A definite integral over an interval represents the net area between the function's graph and the x-axis. Areas above the x-axis contribute positively, while areas below contribute negatively. This net area interpretation is essential for evaluating integrals from graphs.
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Interpreting Areas from Graphs
When given shaded areas on a graph, these represent the absolute values of integrals over subintervals. To find the integral over a larger interval, sum these areas with appropriate signs based on whether the function is above or below the x-axis.
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Properties of Definite Integrals and Limits
The integral from a to 0 can be evaluated by reversing limits: โซโโฐ f(x) dx = -โซโแต f(x) dx. Understanding how to manipulate integral limits and combine subinterval integrals is crucial for solving problems involving integrals over multiple segments.
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Related Practice
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.
ฦ(๐) = ๐ยณ on [โ1, 1]
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Textbook Question
Area versus net area Graph the following functions. Then use geometry (not Riemann sums) to find the area and the net area of the region described.
The region between the graph of y = 1 - |x| and the x-axis, for -2 โค x โค 2
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Textbook Question
Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found.
โซโโต (๐ยฒโ9) d๐
Textbook Question
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
โซ ๐eหฃยฒ d๐
Textbook Question
Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.
โซ ๐ยณ (๐โด + 16)โถ d๐