Textbook Question
Computing areas On the interval [0,2], the graphs of f(x)=x²/3 and g(x)=x²(9−x²)^(-1/2) have similar shapes.
a. Find the area of the region bounded by the graph of f and the x-axis on the interval [0,2].
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 8.2.81a
81. Possible and impossible integrals
Let Iₙ = ∫ xⁿ e⁻ˣ² dx, where n is a nonnegative integer.
a. I₀ = ∫ e⁻ˣ² dx cannot be expressed in terms of elementary functions. Evaluate I₁.
Verified step by step guidance1
Start with the integral definition: \(I_n = \int x^n e^{-x^2} \, dx\). For \(n=1\), we have \(I_1 = \int x e^{-x^2} \, dx\).
Recognize that the integrand \(x e^{-x^2}\) suggests a substitution because the derivative of \(-x^2\) is \(-2x\), which is closely related to the \(x\) term in the integrand.
Use the substitution method: let \(u = -x^2\), then compute \(du = -2x \, dx\), which implies \(x \, dx = -\frac{1}{2} du\).
Rewrite the integral in terms of \(u\): \(I_1 = \int x e^{-x^2} \, dx = \int e^u \left(-\frac{1}{2} du\right) = -\frac{1}{2} \int e^u \, du\).
Integrate with respect to \(u\): \(-\frac{1}{2} \int e^u \, du = -\frac{1}{2} e^u + C\). Finally, substitute back \(u = -x^2\) to express the answer in terms of \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Integration of Functions Involving Exponentials and Polynomials
This concept involves integrating products of polynomial terms and exponential functions, such as xⁿ e⁻ˣ². Techniques like substitution and integration by parts are often used to simplify these integrals, especially when direct antiderivatives are not straightforward.
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Integration by Parts
Integration by parts is a method based on the product rule for differentiation. It transforms the integral of a product of functions into simpler integrals, often reducing the power of polynomials or simplifying exponential terms, which is essential for evaluating integrals like I₁ = ∫ x e⁻ˣ² dx.
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Non-Elementary Integrals and Special Functions
Some integrals, such as ∫ e⁻ˣ² dx, cannot be expressed in terms of elementary functions and are instead represented using special functions like the error function (erf). Recognizing when an integral falls into this category helps in understanding the limitations of standard integration techniques.
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Related Practice
Textbook Question
63. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. If m is a positive integer, then ∫[0 to π] cos^(2m+1)(x) dx = 0.
Textbook Question
65. Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. To evaluate ∫ (4x⁶)/(x⁴ + 3x²) dx, the first step is to find the partial fraction decomposition of the integrand.
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Textbook Question
75. {Use of Tech} Oscillator displacements Suppose a mass on a spring that is slowed by friction has the position function:
s(t) = e⁻ᵗ sin t
a. Graph the position function. At what times does the oscillator pass through the position s = 0?
Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
70. Let f(x) = e^(-x²).
a. Find a Simpson's Rule approximation to the integral from 0 to 3 of e^(-x²) dx using n = 30 subintervals.
Textbook Question
42. Approximating integrals The function f is twice differentiable on (-∞, ∞). Values of f at various points on [0, 20] are given in the table.
a. Approximate ∫(0 to 120) f(x) dx in three way using a left Riemann sum, a right Riemann sum and the Trapezoid Rule