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Chapter 6: Problem 10

\(\int_{0}^{1} e^{-x} d x=\) (A) \(\frac{1}{e}-1\) (B) \(-\frac{1}{e}\) (C) \(1-\frac{1}{e}\) (D) \(\frac{1}{e}\)

Short Answer

Expert verified
(C) \(1-\frac{1}{e}\)

Step by step solution

01

Identify the Integral and Function

The problem requires finding the definite integral of the function \( e^{-x} \) from 0 to 1. It is expressed as \( \int_{0}^{1} e^{-x} \, dx \).
02

Find the Antiderivative

The antiderivative of \( e^{-x} \) is \( -e^{-x} \). This is because the derivative of \( -e^{-x} \) is \( e^{-x} \). We will use this antiderivative to evaluate the definite integral.
03

Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, to evaluate a definite integral, substitute the upper limit and lower limit into the antiderivative. Therefore, evaluate \( -e^{-x} \) from 1 to 0: \[ \left[ -e^{-x} \right]_{0}^{1} = -e^{-1} - (-e^{0}) \]
04

Simplify the Expression

Now simplify the expression: \[ -e^{-1} - (-e^{0}) = -\frac{1}{e} + 1 \] Since \( e^{0} = 1 \), this simplifies to \[ 1 - \frac{1}{e} \].
05

Match with Answer Choices

Now, compare the simplified result \( 1 - \frac{1}{e} \) with the given answer choices. It matches with option (C).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Antiderivative
An antiderivative, also known as an indefinite integral, is a function whose derivative is the given function. For example, if you have the function \( f(x) = e^{-x} \), its antiderivative is \( F(x) = -e^{-x} \). This is because when you differentiate \( -e^{-x} \) with respect to \( x \), you get back \( e^{-x} \). On a broader level, finding the antiderivative is like reverse-engineering differentiation.
  • It helps in setting up the evaluation of definite integrals.
  • An antiderivative is not unique; it can vary by a constant.
In this exercise, identifying \( -e^{-x} \) as the antiderivative is a critical step before applying further calculus concepts like the Fundamental Theorem of Calculus.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is a cornerstone of calculus that connects differentiation and integration. It consists of two parts, but for evaluating definite integrals, the second part is most relevant. This theorem states that if a function \( f(x) \) has an antiderivative \( F(x) \), the integral of \( f(x) \) from \( a \) to \( b \) can be evaluated as:\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
  • It simplifies the computation of definite integrals considerably.
  • It uses antiderivatives to find exact areas under curves.
In our problem, once we identified \( -e^{-x} \) as the antiderivative, we applied this theorem by calculating \( F(b) - F(a) = \left[ -e^{-x} \right]_{0}^{1} \) to get the correct result.
Exponential Functions
Exponential functions, typically of the form \( e^x \) or \( a^x \), play a crucial role in calculus. The function \( e^{-x} \) is a specific type of exponential function where the base \( e \) (Euler's number, approximately 2.718) is raised to the power of \(-x\). Exponential decay is a common context for such functions, where the value decreases rapidly at first and then levels off.
  • Exponential functions have unique properties in calculus, such as appearing in their own derivatives and antiderivatives.
  • They model real-world phenomena like population growth, radioactive decay, and cooling.
In this exercise, understanding how exponential functions integrate and differentiate is key to solving the integral \( \int_{0}^{1} e^{-x} \, dx \). By recognizing the form and behavior of \( e^{-x} \), we efficiently determine the antiderivative and solve the problem.

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Most popular questions from this chapter

The average value of \(\cos x\) over the interval \(\frac{\pi}{3} \leqq x \leqq \frac{\pi}{2}\) is (A) \(\frac{3}{\pi}\) (B) \(\frac{3(2-\sqrt{3})}{\pi}\) (C) \(\frac{3}{2 \pi}\) (D) \(\frac{2}{3 \pi}\)

\(\int_{-1}^{3}|x| d x=\) (A) \(\frac{7}{2}\) (B) 4 (C) \(\frac{9}{2}\) (D) 5

\(\int_{0}^{3} \frac{d t}{\sqrt{4-t}}=\) (A) -2 (B) 4 (C) -1 (D) 2

Choose the Riemann Sum whose limit is the integral \(\int_{1}^{5} x^{3} d x\). (A) \(\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\left(1+\frac{k}{n}\right)^{3} \cdot\left(\frac{4}{n}\right)\right)\) (B) \(\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\left(1+\frac{4 k}{n}\right)^{3} \cdot\left(\frac{1}{n}\right)\right)\) (C) \(\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\left(1+\frac{4 k}{n}\right)^{3} \cdot\left(\frac{4}{n}\right)\right)\) (D) \(\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\left(1+\frac{k}{n}\right)^{3} \cdot\left(\frac{1}{n}\right)\right)\)

Choose the Riemann Sum whose limit is the integral \(\int_{0}^{\pi} \sin (3 x) d x\). (A) \(\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\sin \left(\frac{3 k}{n}\right) \cdot\left(\frac{1}{n}\right)\right)\) (B) \(\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\sin \left(\frac{3 \pi k}{n}\right) \cdot\left(\frac{1}{n}\right)\right)\) (C) \(\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\sin \left(\frac{3 k}{n}\right) \cdot\left(\frac{\pi}{n}\right)\right)\) (D) \(\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\sin \left(\frac{3 \pi k}{n}\right) \cdot\left(\frac{\pi}{n}\right)\right)\)
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