/*! 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 64 Evaluate the following definite ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 64

Evaluate the following definite integrals. $$\int_{0}^{\ln 2}\left(e^{-t} \mathbf{i}+2 e^{2 t} \mathbf{j}-4 e^{t} \mathbf{k}\right) d t$$

Short Answer

Expert verified
Question: Evaluate the definite integral of the vector function $$\vec{F}(t) = e^{-t} \mathbf{i} + 2e^{2t} \mathbf{j} - 4e^t \mathbf{k}$$ within the limits [0, ln(2)]. Answer: $$\frac{1}{2} \mathbf{i} + 3 \mathbf{j} - 4 \mathbf{k}$$

Step by step solution

01

Identify the components of the vector function

The given vector function is: $$ \vec{F}(t) = e^{-t} \mathbf{i} + 2e^{2t} \mathbf{j} - 4e^t \mathbf{k} $$ We have three components in the vector function, corresponding to the i, j, and k directions: $$ f(t)_i = e^{-t} $$ $$ f(t)_j = 2e^{2t} $$ $$ f(t)_k = -4e^t $$
02

Find the definite integral of each component separately

Now we will find the definite integral of each component separately, within the given limits [0, ln(2)]. First component (i-direction): $$ \int_{0}^{\ln 2} e^{-t} dt $$ Second component (j-direction): $$ \int_{0}^{\ln 2} 2e^{2t} dt $$ Third component (k-direction): $$ \int_{0}^{\ln 2} -4e^t dt $$
03

Evaluate the definite integrals

Evaluate the integrals we found in the previous step. First component (i-direction): $$ \int_{0}^{\ln 2} e^{-t} dt = [- e^{-t}]_{0}^{\ln 2} = -e^{-\ln 2} + e^0 = -\frac{1}{2} + 1 = \frac{1}{2} $$ Second component (j-direction): $$ \int_{0}^{\ln 2} 2e^{2t} dt = [e^{2t}]_{0}^{\ln 2} = e^{2\ln 2} - e^0 = 2^2 - 1 = 3 $$ Third component (k-direction): $$ \int_{0}^{\ln 2} -4e^t dt = -4[e^t]_{0}^{\ln 2} = -4(e^{\ln 2} - e^0) = -4(2 - 1) = -4 $$
04

Combine the results and form the final vector after integrating

Now that we have the definite integrals of each component, we can combine them to form the final vector. The final vector after integrating is: $$ \frac{1}{2} \mathbf{i} + 3 \mathbf{j} - 4 \mathbf{k} $$

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!

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

Use the formula in Exercise 79 to find the (least) distance between the given point \(Q\) and line \(\mathbf{r}\). $$Q(5,6,1) ; \mathbf{r}(t)=\langle 1+3 t, 3-4 t, t+1\rangle$$

Compute \(\mathbf{r}^{\prime \prime}(t)\) and \(\mathbf{r}^{\prime \prime \prime}(t)\) for the following functions. $$\mathbf{r}(t)=\left\langle 3 t^{12}-t^{2}, t^{8}+t^{3}, t^{-4}-2\right\rangle$$

Evaluate the following definite integrals. $$\int_{0}^{\pi / 4}\left(\sec ^{2} t \mathbf{i}-2 \cos t \mathbf{j}-\mathbf{k}\right) d t$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The vectors \(\mathbf{r}(t)\) and \(\mathbf{r}^{\prime}(t)\) are parallel for all values of \(t\) in the domain. b. The curve described by the function \(\mathbf{r}(t)=\left\langle t, t^{2}-2 t, \cos \pi t\right\rangle\) is smooth, for \(-\infty

An ant walks due east at a constant speed of \(2 \mathrm{mi} / \mathrm{hr}\) on a sheet of paper that rests on a table. Suddenly, the sheet of paper starts moving southeast at \(\sqrt{2} \mathrm{mi} / \mathrm{hr} .\) Describe the motion of the ant relative to the table.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

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.