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 8.5.65a
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.
Verified step by step guidance1
First, examine the integrand \( \frac{4x^{6}}{x^{4} + 3x^{2}} \). Notice that both numerator and denominator are polynomials in terms of \( x \).
Check the degree of the numerator and denominator: the numerator has degree 6, and the denominator has degree 4. Since the degree of the numerator is higher, consider simplifying the integrand by polynomial division before attempting partial fraction decomposition.
Perform polynomial division of \( 4x^{6} \) by \( x^{4} + 3x^{2} \) to rewrite the integrand as a polynomial plus a proper rational function (where the numerator degree is less than the denominator degree).
After polynomial division, if the remaining rational function has a denominator that can be factored into linear or irreducible quadratic factors, then partial fraction decomposition can be applied to that part.
Therefore, the first step is not to directly find the partial fraction decomposition of the original integrand, but to simplify it first by polynomial division. This shows that the statement is false.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Partial Fraction Decomposition
Partial fraction decomposition is a technique used to break down rational functions into simpler fractions that are easier to integrate. It applies when the integrand is a ratio of polynomials where the degree of the numerator is less than the degree of the denominator. This method is not suitable if the numerator's degree is equal to or greater than the denominator's.
Recommended video:
10:07
Partial Fraction Decomposition: Distinct Linear Factors
Polynomial Degree Comparison
Before applying partial fraction decomposition, compare the degrees of the numerator and denominator polynomials. If the numerator's degree is higher or equal, perform polynomial division first to rewrite the integrand as a polynomial plus a proper fraction. This step ensures the integrand fits the form required for partial fractions.
Recommended video:
07:00Taylor Polynomials
Integration of Rational Functions
Integrating rational functions often involves simplifying the integrand through algebraic manipulation such as polynomial division or partial fractions. Recognizing the appropriate method depends on the form of the integrand, which guides the integration strategy and helps avoid unnecessary or incorrect steps.
Recommended video:
6:04Intro to Rational Functions
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
A piece of wood paneling must be cut in the shape shown in the figure.
The coordinates of several points on its curved surface are also shown (with units of inches).
a. Estimate the surface area of the paneling using the Trapezoid Rule.
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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.
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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.
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Textbook Question
66–71. {Use of Tech} Estimating error Refer to Theorem 8.1 in the following exercises.
69. Let f(x) = sin(eˣ).
a. Find a Trapezoid Rule approximation to ∫[0 to 1] sin(eˣ) dx using n = 40 subintervals.
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