Textbook Question
49. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample:
c. ∫ v du = u·v - ∫ u dv
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Ch. 8 - Integration Techniques
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 8, Problem 8.2.82b
82. A family of exponentials The curves y = x * e^(-a * x) are shown in the figure for a = 1, 2, and 3.
b. Find the area of the region bounded by y = x * e^(-a * x) and the x-axis on the interval [0, 4], where a > 0.![Graph showing three curves of y = x * e^(-a * x) for a = 1, 2, and 3, decreasing and approaching zero on [0,4].](https://static.studychannel.pearsonprd.tech/courses/calculus/thumbnails/a3b793f9-f31f-40a3-a153-ba6be7b8c3a3)
Verified step by step guidance1
Identify the function to integrate: the area under the curve y = x e^{-a x} from x = 0 to x = 4 is given by the definite integral \( \int_0^4 x e^{-a x} \, dx \), where \( a > 0 \).
Set up the integral explicitly: \( \text{Area} = \int_0^4 x e^{-a x} \, dx \). This integral represents the area bounded by the curve and the x-axis over the interval [0,4].
Use integration by parts to solve the integral. Let \( u = x \) and \( dv = e^{-a x} dx \). Then, compute \( du = dx \) and \( v = \int e^{-a x} dx = -\frac{1}{a} e^{-a x} \).
Apply the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). Substitute the expressions to get \( \int_0^4 x e^{-a x} dx = \left. -\frac{x}{a} e^{-a x} \right|_0^4 + \frac{1}{a} \int_0^4 e^{-a x} dx \).
Evaluate the remaining integral \( \int_0^4 e^{-a x} dx \), which is \( \left. -\frac{1}{a} e^{-a x} \right|_0^4 \). Then, combine all terms to express the area in terms of \( a \) without calculating the numerical value.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral and Area Under a Curve
The definite integral of a function over an interval represents the net area between the curve and the x-axis. For positive functions, this corresponds to the actual area bounded by the curve, the x-axis, and the vertical lines at the interval endpoints. Calculating this integral gives the total area under the curve on the specified interval.
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Definition of the Definite Integral
Integration by Parts
Integration by parts is a technique used to integrate products of functions. It is based on the product rule for differentiation and is especially useful when integrating expressions like x * e^(-ax). The formula is ∫u dv = uv - ∫v du, where u and dv are parts of the original integrand chosen to simplify the integral.
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Exponential Decay and Parameter Effects
The function y = x * e^(-ax) combines a linear term and an exponential decay term. The parameter 'a' controls the rate of decay: larger 'a' values cause the function to decrease more rapidly. Understanding how 'a' affects the shape and area under the curve is crucial for evaluating the integral and interpreting the graph.
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Related Practice
Textbook Question
45–48. {Use of Tech} Trapezoid Rule and Simpson’s Rule Consider the following integrals and the given values of n.
46. ∫(0 to 2) x⁴ dx; n = 30
c. Compute the absolute errors in the Trapezoid Rule and Simpson’s Rule with 2n subintervals.
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Textbook Question
2. Give an example of each of the following.
b. A repeated linear factor
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Textbook Question
Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with m ≠ n.
c.
π
∫ sin(mx) cos(nx) dx = 0, when |m + n| is even
0
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Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
67. Let f(x) = √(x³ + 1).
b. Calculate f''(x).
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Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
71. Let f(x) = √(sin x).
b. Find an upper bound on the absolute error in the estimate from part (a) using Theorem 8.1. (Hint: |f''''(x)| ≤ 1 on [1,2].)
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