Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Riemann Sum
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Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
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![**Instructions for Calculus Problem**
**Objective:** Show that the value of the integral
\[ \int_{0}^{\infty} e^{-2t/RC} \, dt \]
is equal to
\[ \frac{1}{2} \tau. \]
**Context:**
In this problem, we are working with an exponential decay function often encountered in electrical engineering, specifically with resistor-capacitor (RC) circuits. The task is to evaluate the given integral and show that its solution equals \( \frac{1}{2} \tau \), where \( \tau \) represents the time constant of the RC circuit.
**Steps to Solve:**
1. **Understand the Function:** Recognize that \( e^{-2t/RC} \) is an exponential decay function with respect to time \( t \).
2. **Set Up the Integral:**
The integral we need to solve is:
\[
\int_{0}^{\infty} e^{-2t/RC} \, dt
\]
3. **Solve the Integral:**
- First, identify the integral of the exponential function.
- Determine the indefinite integral:
\[
\int e^{-2t/RC} \, dt
\]
- Use substitution if necessary to simplify the exponent.
- Evaluate the definite integral from \( t = 0 \) to \( t = \infty \).
4. **Apply the Limits:**
- Substitute the limits into the antiderivative to complete the evaluation.
5. **Compare with \( \frac{1}{2} \tau \):**
- Show that the resulting expression equates to \( \frac{1}{2} \tau \).
**Explanation:**
The problem involves exponential decay, characterized by the integral of an exponential function. Such integrals often arise in calculating total charge or energy in systems described by similar decay processes, including RC circuits where \( \tau = RC \).
Understanding and solving this problem helps in grasping the mathematical descriptions of physical processes involving decay and systems' response over time.
Feel free to use pen and paper to follow along with the integration process or use a computational tool to verify your results.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb6f89a01-a0d2-4d6f-ab48-f2d74370b2c9%2F552e3c5c-114b-4357-843b-9544f0f89310%2Fiw3k8fah_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Instructions for Calculus Problem**
**Objective:** Show that the value of the integral
\[ \int_{0}^{\infty} e^{-2t/RC} \, dt \]
is equal to
\[ \frac{1}{2} \tau. \]
**Context:**
In this problem, we are working with an exponential decay function often encountered in electrical engineering, specifically with resistor-capacitor (RC) circuits. The task is to evaluate the given integral and show that its solution equals \( \frac{1}{2} \tau \), where \( \tau \) represents the time constant of the RC circuit.
**Steps to Solve:**
1. **Understand the Function:** Recognize that \( e^{-2t/RC} \) is an exponential decay function with respect to time \( t \).
2. **Set Up the Integral:**
The integral we need to solve is:
\[
\int_{0}^{\infty} e^{-2t/RC} \, dt
\]
3. **Solve the Integral:**
- First, identify the integral of the exponential function.
- Determine the indefinite integral:
\[
\int e^{-2t/RC} \, dt
\]
- Use substitution if necessary to simplify the exponent.
- Evaluate the definite integral from \( t = 0 \) to \( t = \infty \).
4. **Apply the Limits:**
- Substitute the limits into the antiderivative to complete the evaluation.
5. **Compare with \( \frac{1}{2} \tau \):**
- Show that the resulting expression equates to \( \frac{1}{2} \tau \).
**Explanation:**
The problem involves exponential decay, characterized by the integral of an exponential function. Such integrals often arise in calculating total charge or energy in systems described by similar decay processes, including RC circuits where \( \tau = RC \).
Understanding and solving this problem helps in grasping the mathematical descriptions of physical processes involving decay and systems' response over time.
Feel free to use pen and paper to follow along with the integration process or use a computational tool to verify your results.
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