Textbook Question
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ θ cos(πθ) dθ
Ch. 8 - Techniques of Integration
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 8, Problem 8.8.83
Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
83. Find the area of the region.
Verified step by step guidance1
Identify the region whose area you want to find. Here, it is the area in the first quadrant bounded by the curve \(y = e^{-x}\) and the x-axis.
Set up the definite integral to find the area. Since the region is in the first quadrant and bounded by the x-axis, the limits of integration for \(x\) are from 0 to \(\infty\). The area \(A\) can be expressed as \(A = \int_0^{\infty} e^{-x} \, dx\).
Recall the integral formula for the exponential function: \(\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C\) for \(a \neq 0\). In this case, \(a = -1\).
Evaluate the improper integral by taking the limit as the upper bound approaches infinity: \(A = \lim_{t \to \infty} \int_0^t e^{-x} \, dx\).
Compute the definite integral using the antiderivative and then apply the limit to find the area.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral as Area Under a Curve
The definite integral of a function over an interval represents the net area between the curve and the x-axis within that interval. For positive functions, this corresponds to the actual area bounded by the curve, the x-axis, and the vertical limits.
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05:43
Definition of the Definite Integral
Improper Integrals and Infinite Limits
When the region extends infinitely, such as from a finite point to infinity, the integral is called an improper integral. Evaluating it involves taking a limit as the upper bound approaches infinity to determine if the area converges to a finite value.
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11:11Improper Integrals: Infinite Intervals
Exponential Decay Function y = e^(-x)
The function y = e^(-x) is an exponential decay curve that approaches zero as x approaches infinity. Its properties ensure that the area under the curve from zero to infinity converges, making it suitable for calculating finite areas over infinite intervals.
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03:39Integrals of Natural Exponential Functions (e^x)
Related Practice
Textbook Question
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (s⁴ + 81) / (s(s² + 9)²) ds
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Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (dx / ((2x + 1)√(4x + 4x²)))
Textbook Question
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (2^(√y) dy) / 2√y
Textbook Question
Expand the quotients in Exercises 1–8 by partial fractions.
z / (z³ - z² - 6z)
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Textbook Question
In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.
∫ from π to ∞ of ((1 + sin x) / x² dx)
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