Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. ∫(3/(x² + 4)) dx = ∫(3/x²) dx + ∫(3/4) dx.
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 8.9.91a
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.
Verified step by step guidance1
Identify the region R bounded by the curve \(y = e^{-a \cdot x}\) and the x-axis on the interval \([b, \infty)\). Since the curve is above the x-axis for \(a > 0\), the area can be found by integrating the function from \(x = b\) to \(x = \infty\).
Set up the integral for the area \(A(a,b)\) as:
\(A(a,b) = \int_{b}^{\infty} e^{-a \cdot x} \, dx\)
Recall the integral formula for the exponential function:
\(\int e^{kx} \, dx = \frac{1}{k} e^{kx} + C\), where \(k\) is a constant. Here, \(k = -a\).
Evaluate the definite integral by applying the limits from \(b\) to \(\infty\):
\(A(a,b) = \left[ \frac{e^{-a \cdot x}}{-a} \right]_{b}^{\infty}\)
Calculate the limit as \(x \to \infty\) of \(e^{-a \cdot x}\), which approaches 0 for \(a > 0\), and then substitute \(x = b\) to express the area \(A(a,b)\) in terms of \(a\) and \(b\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integrals and Area Under a Curve
The area under a curve between two points is found using definite integrals. For a function f(x), the area from x = b to x = c is the integral of f(x) dx over [b, c]. When the upper limit is infinity, improper integrals are used to evaluate the area.
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05:43
Definition of the Definite Integral
Exponential Functions and Their Properties
Exponential functions like y = e^(-a·x) decay rapidly as x increases when a > 0. Understanding their behavior helps in setting up integrals and evaluating limits, especially for infinite intervals where the function approaches zero.
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06:21Properties of Functions
Improper Integrals and Convergence
When integrating over an infinite interval, the integral is called improper. To find the area, one must evaluate the limit of the integral as the upper bound approaches infinity and verify that this limit converges to a finite value.
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11:11Improper Integrals: Infinite Intervals
Related Practice
Textbook Question
Arc length of a parabola Let L(c) be the length of the parabola f(x) = x² from x = 0 to x = c, where c ≥ 0 is a constant.
a. Find an expression for L.
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Textbook Question
60. Two Methods
a. Evaluate ∫(x · ln(x²)) dx using the substitution u = x² and evaluating ∫(ln(u)) du.
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
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
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.