/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 12 Evaluate the integrals. $$ \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 12

Evaluate the integrals. $$ \int_{-1}^{0} 3 e^{x} d x $$

Short Answer

Expert verified
The definite integral \(\int_{-1}^{0} 3 e^{x} dx\) can be evaluated by first finding the antiderivative of the function, which is \(F(x) = 3e^x + C\). Applying the fundamental theorem of calculus, we get the result \(3 - \frac{3}{e}\).

Step by step solution

01

Find the antiderivative of the function

To find the antiderivative of the function 3e^x, we need to find a function whose derivative is equal to 3e^x. Recall the differentiation rules: \(\\ \frac{d}{dx}(e^x) = e^x\\ \frac{d}{dx}(c \cdot f(x)) = c \cdot \frac{d f(x)}{d x} \\ \) Applying these rules to this problem, the antiderivative of 3e^x is: \[ F(x) = 3e^x + C \] where C is the constant of integration.
02

Apply the fundamental theorem of calculus

The fundamental theorem of calculus allows us to evaluate the definite integral of a function by finding the difference in the antiderivative at the given limits: \[ \int_{-1}^{0} 3 e^{x} d x = F(0) - F(-1) \]
03

Evaluate the antiderivative at the given limits

Now, substitute the limits of integration into the antiderivative function: \[ F(0) = 3e^0 + C = 3 + C \\ F(-1) = 3e^{-1} + C = \frac{3}{e} + C \] Since we are evaluating a definite integral, the constant of integration, C, will cancel out in the final result.
04

Substitute the antiderivative values into the fundamental theorem of calculus

Using the values from step 3, substitute them into the fundamental theorem of calculus expression: \[ \int_{-1}^{0} 3 e^{x} d x = F(0) - F(-1) = (3 + C) - \left(\frac{3}{e} + C\right) \] Cancel out the C terms and compute the definite integral: \[ \int_{-1}^{0} 3 e^{x} d x = 3 - \frac{3}{e} \] So the value of the definite integral is \(3 - \frac{3}{e}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Antiderivatives
Antiderivatives are the reverse of differentiation. If you have a function like 3e^x, its antiderivative is a function whose derivative gives us back 3e^x. To find it, you use the rules of differentiation in reverse.

For example, since the derivative of e^x is e^x, the antiderivative of 3e^x is simply 3e^x, but with an added constant of integration, C. This constant represents all possible vertical shifts of the antiderivative function, since the derivative of any constant is zero.
Fundamental Theorem of Calculus
The fundamental theorem of calculus is the bridge between differentiation and integration. It states that if you want to evaluate the definite integral of a function over an interval, you can do this by finding the difference in its antiderivative at the endpoints of the interval.

This theorem comes in two main parts: the first part ensures that the antiderivative can be used to calculate definite integrals, and the second part relates the derivative of the antiderivative to the original function.
Integration Limits
Integration limits define the interval over which you're integrating. In the exercise, the lower limit is -1 and the upper limit is 0. These limits are the endpoints used in the fundamental theorem of calculus, showing where to evaluate the antiderivative.

Remember, definite integrals are not affected by the constant of integration because it cancels out when you're subtracting the antiderivative evaluated at the upper limit from the lower limit.
e Exponential Function
The exponential function with base e is extraordinarily important in calculus due to its unique properties. For instance, its derivative is the same as the function itself, d/dx(e^x) = e^x.

In this problem, we deal with e^x and its reciprocal e^{-1} or 1/e. The presence of the e function in integrals often simplifies things because its antiderivative is so straightforward, leading to elegant solutions in many definite integral problems.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(Compare Exercise 59 in Section 13.3.) The total number of wiretaps authorized each year by U.S. state and federal courts from 1990 to 2010 can be approximated by $$ w(t)=820 e^{0.051 t} \quad(0 \leq t \leq 20) $$ \((t\) is time in years since the start of 1990\() .^{41}\) Compute \(\int_{0}^{15} w(t) d t\). (Round your answer to the nearest 10.) Interpret the answer.

Calculate the total area of the regions described. Do not count area beneath the \(x\) -axis as negative. HINT [See Example 6.] Bounded by the line \(y=2 x\), the \(x\) -axis, and the lines \(x=1\) and \(x=2\)

Evaluate the integrals. $$ \int_{0}^{2} \frac{3 x}{x^{2}+2} d x $$

What is 鈥渄efinite鈥� about the definite integral?

Evaluate the integrals. $$ \int_{2}^{3}\left(x+\frac{1}{x}\right) d x $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.