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Chapter 4: Problem 54

Evaluate. $$ \int_{a}^{b}-e^{t} d t $$

Short Answer

Expert verified
The integral evaluates to \( e^a - e^b \).

Step by step solution

01

Identify the Structure of the Integral

The integral to evaluate is a definite integral, which means it has limits of integration (from \( a \) to \( b \)) and involves the exponential function \( -e^t \). We need to perform integration followed by substitution of the limits to solve it.
02

Integrate the Function

To find the antiderivative of \( -e^t \), we use the basic rule that the integral of \( e^t \) is \( e^t \) itself. Thus, the integral of \( -e^t \) is \(-e^t\).
03

Apply the Fundamental Theorem of Calculus

According to the Fundamental Theorem of Calculus, the value of the definite integral \( \int_{a}^{b} -e^t \, dt \) is the antiderivative evaluated at \( b \) minus the antiderivative evaluated at \( a \). Thus, the result of this integral is:\[-e^b - (-e^a) = -e^b + e^a.\]
04

Simplify the Expression

The expression \(-e^b + e^a\) is already in the simplest form. This is the result of the definite integral after applying the limits.

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Key Concepts

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

Exponential Function
Exponential functions, such as \( e^t \), are a fundamental part of mathematics and calculus. In an exponential function, a constant base (\( e \), the Euler's number, which is approximately 2.71828) is raised to the power of a variable. Exponential functions grow very rapidly and frequently appear in various applications including natural growth processes and compound interest calculations.
  • The most common exponential function is expressed as \( e^t \), where \( e \) is the base and \( t \) is the exponent.
  • These functions are characterized by their continuous and smooth curvature as well as their ability to model growth at a consistent rate.
  • In this exercise, we are working with the function \( -e^t \), which is simply the negative form of the standard exponential function.
Understanding and working with exponential functions is crucial, particularly in calculus where they are often integrated and differentiated to solve complex problems.
Antiderivative
An antiderivative, or the indefinite integral, is the reverse process of differentiation. It allows us to reconstruct a function from its derivative. Knowing how to find an antiderivative is essential for solving integration problems.
  • For example, when you integrate a function like \( e^t \), its antiderivative is \( e^t \) itself because the derivative of \( e^t \) is \( e^t \).
  • In the exercise, we are evaluating the antiderivative of \( -e^t \), which is \(-e^t\). This follows from the linearity of integration, where a constant multiplier like \(-1\) is factored out.
      • Recognizing patterns in differentiation and knowing basic integration rules like this helps in handling more complex calculus problems.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is a critical bridge connecting differentiation and integration. It states that if a function is continuous on a closed interval, then the definite integral over that interval is connected to its antiderivative.
  • Here's how it works: If you have a function \( F(t) \) that is an antiderivative of \( f(t) \), then the definite integral of \( f(t) \) from \( a \) to \( b \) is \( F(b) - F(a) \).
  • In our example, given the antiderivative \(-e^t\), we evaluate \(-e^b\) and \(-e^a\), and subtract these results to obtain \(-e^b + e^a\).
The theorem provides a straightforward method to evaluate integrals and is key to solving real-world problems efficiently by transforming problems of finding areas into simple subtraction tasks.

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Most popular questions from this chapter

Explain the error that has been made in each. $$ \begin{aligned} \int_{1}^{2}\left(\ln x-e^{x}\right) d x &=\left[\frac{1}{x}-e^{x}\right]_{1}^{2} \\ &=\left(\frac{1}{2}-e^{2}\right)-\left(1-e^{1}\right) \\ &=e-e^{2}-\frac{1}{2} \end{aligned} $$

Use geometry to evaluate each definite integral. \(\int_{2}^{6} 3 d x\)

Solve each integral. Each can be found using rules developed in this section, but some algebra may be required. $$ \int \frac{t^{3}+8}{t+2} d t $$

Evaluate. $$ \int x^{2} \sqrt{x^{3}+1} d x $$

Evaluate. $$ \int_{0}^{\sqrt{7}} 7 x \sqrt[3]{1+x^{2}} d x $$
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