/*! 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 59 Compute the indefinite integral ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 59

Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=\left\langle t^{4}-3 t, 2 t-1,10\right\rangle$$

Short Answer

Expert verified
Answer: The indefinite integral of the given vector-valued function is \(\mathbf{R}(t) = \left\langle \frac{t^5}{5} - \frac{3t^2}{2} + C_1, t^2 - t + C_2, 10t + C_3 \right\rangle\), where \(C_1\), \(C_2\), and \(C_3\) are constants of integration.

Step by step solution

01

Separate the components of the vector

We can represent the given vector function, \(\mathbf{r}(t)\), as three separate scalar functions: $$f(t) = t^4 - 3t,$$ $$g(t) = 2t - 1,$$ $$h(t) = 10.$$
02

Integrate each component

Now, we will find the indefinite integral of each scalar function, \(f(t)\), \(g(t)\), and \(h(t)\): $$F(t) = \int (t^4 - 3t) dt,$$ $$G(t) = \int (2t - 1) dt,$$ $$H(t) = \int 10 dt.$$
03

Find the antiderivative of each component

Now, we will find the antiderivative of each of these scalar functions: $$F(t) = \frac{t^5}{5} - \frac{3t^2}{2} + C_1,$$ $$G(t) = t^2 - t + C_2,$$ $$H(t) = 10t + C_3,$$ where \(C_1\), \(C_2\), and \(C_3\) are constants of integration.
04

Combine the components

Finally, we will combine the results of our calculations above to form the new vector-valued function, \(\mathbf{R}(t)\): $$\mathbf{R}(t) = \left\langle \frac{t^5}{5} - \frac{3t^2}{2} + C_1, t^2 - t + C_2, 10t + C_3 \right\rangle.$$ This is the indefinite integral of the given vector-valued function, \(\mathbf{r}(t)\).

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.

Indefinite Integral
When solving problems in calculus, indefinite integrals play a crucial role. An indefinite integral is the reverse operation of differentiation. It helps us find a function whose derivative is the given function. This process is also known as integration without limits.
For real-valued functions, the indefinite integral takes the form:
  • \( \int f(x) \, dx = F(x) + C \)
Here, \(F(x)\) is the antiderivative of \(f(x)\), and \(C\) represents the constant of integration. Indefinite integrals help us discover families of functions that vary by this constant. This is important because the process of differentiation loses information about constant terms in the function, which integration restores.
When dealing with vector-valued functions, we apply these integration concepts component-wise, meaning each dimension of the vector is integrated separately.
Antiderivative
The concept of an antiderivative is foundational to understanding indefinite integrals. The antiderivative works as the inverse of differentiation. It reconstructs a function from its derivative. Another way to describe it is that if \(F(x)\) is an antiderivative of \(f(x)\), then \(F'(x) = f(x)\).
Let's explore how to find one. Consider \(f(x)\) as your function, and look for some function \(F(x)\) that satisfies the following:
  • \(\frac{d}{dx}[F(x)] = f(x)\)
Remember that there are infinitely many antiderivatives for any given function. All differ by a constant. Thus, the notation \(F(x) + C\) captures this infinite family, where \(C\) is any real number. If you differentiate \(F(x) + C\), you still get \(f(x)\), as the derivative of a constant is zero.
In vector-valued functions, an antiderivative individually handles each component of the vector. This keeps each part's distinct identity while assembling the general solution.
Vector-Valued Function
Vector-valued functions expand the idea of a regular function to include output in vector form. In essence, they map a single input value, like \(t\), into a multi-dimensional vector.
These functions are represented as:
  • \( \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle \)
Where \(f(t)\), \(g(t)\), and \(h(t)\) are the component functions.
Understanding vector-valued functions involves operating on their components separately. They are embedded in contexts like physics and engineering, where motions or forces have directions represented by vectors.
To compute an indefinite integral for such functions, integrate each of the component functions one by one. Once integrated, combine the results to form the final vector, which will express the integrated relation. This step-by-step component-focused approach makes it easier to manage multifaceted problems common in real-world applications.

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

Compute the following derivatives. $$\frac{d}{d t}\left(\left(t^{3} \mathbf{i}-2 t \mathbf{j}-2 \mathbf{k}\right) \times\left(t \mathbf{i}-t^{2} \mathbf{j}-t^{3} \mathbf{k}\right)\right)$$

Find the function \(\mathrm{r}\) that satisfies the given conditions. $$\mathbf{r}^{\prime}(t)=\left\langle e^{t}, \sin t, \sec ^{2} t\right\rangle ; \quad \mathbf{r}(0)=\langle 2,2,2\rangle$$

Let \(\mathbf{u}(t)=\left\langle 1, t, t^{2}\right\rangle, \mathbf{v}(t)=\left\langle t^{2},-2 t, 1\right\rangle\) and \(g(t)=2 \sqrt{t}\). Compute the derivatives of the following functions. $$\mathbf{u}(t) \cdot \mathbf{v}(t)$$

Nonuniform straight-line motion Consider the motion of an object given by the position function $$\mathbf{r}(t)=f(t)\langle a, b, c\rangle+\left\langle x_{0}, y_{0}, z_{0}\right\rangle, \quad \text { for } t \geq 0$$ where \(a, b, c, x_{0}, y_{0},\) and \(z_{0}\) are constants, and \(f\) is a differentiable scalar function, for \(t \geq 0\) a. Explain why \(r\) describes motion along a line. b. Find the velocity function. In general, is the velocity constant in magnitude or direction along the path?

A projectile (such as a bascball or a cannonball) launched from the origin with an initial horizontal velocity \(u_{0}\) and an initial vertical velocity \(v_{0}\) moves in a parabolic trajectory given by $$\mathbf{r}(t)=\left\langle u_{0} t_{1}-\frac{1}{2} g t^{2}+v_{0} t\right\rangle, \quad \text { for } t \geq 0$$ where air resistance is neglected and \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity (see Section 14.3 ). a. Let \(u_{0}=20 \mathrm{m} / \mathrm{s}\) and \(v_{0}=25 \mathrm{m} / \mathrm{s} .\) Assuming the projectile is launched over horizontal ground, at what time does it return to Earth? b. Find the integral that gives the length of the trajectory from launch to landing. c. Evaluate the integral in part (b) by first making the change of variables \(u=-g t+v_{0} .\) The resulting integral is evaluated either by making a second change of variables or by using a calculator. What is the length of the trajectory? d. How far does the projectile land from its launch site?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

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.