Textbook Question
91. [Use of Tech] Regions bounded by exponentials Let a > 0 and let R be the region bounded by the graph of y = e^(-a·x) and the x-axis
on the interval [b, ∞).
a. Find A(a,b), the area of R as a function of a and b.
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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 8.8.70a
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.
Verified step by step guidance1
Step 1: Recall Simpson's Rule formula for approximating integrals: \( \int_{a}^{b} f(x) dx \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + \dots + 4f(x_{n-1}) + f(x_n) \right] \), where \( h = \frac{b-a}{n} \) and \( x_i = a + i \cdot h \).
Step 2: Identify the parameters for the problem. Here, \( f(x) = e^{-x^2} \), \( a = 0 \), \( b = 3 \), and \( n = 30 \). Calculate \( h \) using \( h = \frac{b-a}{n} \), which gives \( h = \frac{3-0}{30} = 0.1 \).
Step 3: Generate the \( x_i \) values. These are the points \( x_0, x_1, x_2, \dots, x_{30} \), where \( x_i = a + i \cdot h \). For example, \( x_0 = 0 \), \( x_1 = 0.1 \), \( x_2 = 0.2 \), and so on up to \( x_{30} = 3 \).
Step 4: Evaluate \( f(x_i) \) for each \( x_i \). Compute \( f(x_i) = e^{-x_i^2} \) for all \( x_i \) values. For example, \( f(x_0) = e^{-(0)^2} = 1 \), \( f(x_1) = e^{-(0.1)^2} \), and so on.
Step 5: Apply Simpson's Rule formula. Substitute the \( f(x_i) \) values into the formula, ensuring the coefficients alternate between 4 and 2 for interior points, with \( f(x_0) \) and \( f(x_{30}) \) having coefficients of 1. Multiply by \( \frac{h}{3} \) to complete the approximation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Simpson's Rule
Simpson's Rule is a numerical method for approximating the definite integral of a function. It uses parabolic segments to estimate the area under the curve, providing a more accurate approximation than methods like the Trapezoidal Rule. The formula involves evaluating the function at evenly spaced points and applying weights to these values, making it particularly effective for smooth functions.
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Power Rules
Definite Integral
A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The result of a definite integral is a number that quantifies the total accumulation of the function's values across the interval.
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Error Estimation
Error estimation in numerical integration involves determining how closely the approximation of an integral matches the actual value. Theorem 8.1 likely provides a framework for estimating the error associated with Simpson's Rule, which can depend on factors such as the number of subintervals and the behavior of the function being integrated. Understanding error helps in assessing the reliability of the numerical approximation.
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04:57Determining Error and Relative Error
Related Practice
Textbook Question
Gamma function The gamma function is defined by Γ(p) = ∫ from 0 to ∞ of x^(p-1) e^(-x) dx, for p not equal to zero or a negative integer.
a. Use the reduction formula ∫ from 0 to ∞ of x^p e^(-x) dx = p ∫ from 0 to ∞ of x^(p-1) e^(-x) dx for p = 1, 2, 3, ...
to show that Γ(p + 1) = p! (p factorial).
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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].
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
41-44. {Use of Tech} Nonuniform grids
Use the indicated methods to solve the following problems with nonuniform grids.
41. A curling iron is plugged into an outlet at time t = 0. Its temperature T in degrees Fahrenheit, assumed to be a continuous function that is strictly increasing and concave down on 0 ≤ t ≤ 120, is given at various times (in seconds) in the table.
a. Approximate (1/120)∫(0 to 120)T(t)dt in three ways using a left Riemann sum, using a right Riemann sum and using the Trapezoid Rule
Interpret the value of (1/120)∫(0 to 120)T(t)dt in the context of this problem.
Textbook Question
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.