Domanda del libro di testo
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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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 8, Problema 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)
Guida verificata passo dopo passo1
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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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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Percorso guidato
05:43
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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06:18Integration by Parts for Definite Integrals
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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09:29Exponential Growth & Decay
Pratica correlata
Domanda del libro di testo
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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Domanda del libro di testo
2. Give an example of each of the following.
b. A repeated linear factor
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Domanda del libro di testo
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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Domanda del libro di testo
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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Domanda del libro di testo
59. Two Methods
b. Evaluate ∫(x / √(x + 1)) dx using substitution.
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