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Chapter 14: Problem 31

Evaluate the indefinite integral. $$\int(3 \mathbf{i}+4 t \mathbf{j}) d t$$

Short Answer

Expert verified
The integral is \(3t \mathbf{i} + 2t^2 \mathbf{j} + \mathbf{C}\).

Step by step solution

01

Recognize the Components

Observe that the integral is a vector function with two components: a constant component, \(3 \mathbf{i}\), and a function of \(t\), \(4t \mathbf{j}\).
02

Integrate Each Component Separately

Integrate the constant component, \(3 \mathbf{i}\), with respect to \(t\). The integral of a constant is the constant multiplied by the variable of integration, so we get \(3t \mathbf{i}\). For the \(4t \mathbf{j}\) component, integrate \(4t\) to get \(2t^2 \mathbf{j}\).
03

Add the Constant of Integration

Since this is an indefinite integral, add a constant of integration, \(\mathbf{C}\). Combine the integrated components: \(\int(3 \mathbf{i}+4 t \mathbf{j}) \, dt = 3t \mathbf{i} + 2t^2 \mathbf{j} + \mathbf{C}\).

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Key Concepts

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

Vector Calculus
Vector calculus is a branch of mathematics that deals with vector fields and the operations that you can perform on them. It combines aspects of calculus and linear algebra and often extends them to analyze multidimensional spaces. The given integral involves a vector function which has a vector along the
  • i-component as a constant.
  • j-component depends on the variable t.
In this problem, each component of the vector is treated separately, which is a key technique in vector calculus. We're looking at how a function behaves over a vector space through integration, a concept that is quite common in fields like physics and engineering.
Constant of Integration
When you solve an indefinite integral, there's always an unknown or arbitrary constant added to the solution, known as the constant of integration. This constant arises because indefinite integration essentially reverses the process of differentiation. Since the differentiation of a constant is zero, different functions that only differ by a constant will give the same derivative. Thus,
  • for every indefinite integral, we add a constant to capture all possible solutions.
  • In vector calculus, this constant becomes a vector itself, denoted as \(\mathbf{C}\) in our case above.
It ensures completeness of the solution, ensuring that every possible antiderivative is accounted for, not just a single one.
Integration by Components
Integration by components is not as commonly discussed as its counterpart, integration by parts, but it plays a significant role in tasks like the one provided, where functions are expressed as sum or difference of separate, individual terms. Each term is treated independently, integrating them separately, then combining their results.
  • This method simplifies complex integrals by breaking them down.
  • In the exercise, the vector function is treated as two separate parts: the \(3\mathbf{i}\) and \(4t\mathbf{j}\).
Doing this allows you to apply basic integration rules to each component individually, leading to a simpler and more efficient path to the solution.

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Most popular questions from this chapter

At time \(t=0\) a baseball that is \(5 \mathrm{ft}\) above the ground is hit with a bat. The ball leaves the bat with a speed of \(80 \mathrm{ft} / \mathrm{s}\) at an angle of \(30^{\circ}\) above the horizontal. (a) How long will it take for the baseball to hit the ground? Express your answer to the nearest hundredth of a second. (b) Use the result in part (a) to find the horizontal distance traveled by the ball. Express your answer to the nearest tenth of a foot.

We assume that \(s\) is an arc length parameter for a smooth vector-valued function \(\mathbf{r}(s)\) in 3 -space and that \(d\) T/ds and \(d\) N/ds exist at each point on the curve. This implies that \(d\) B/ds exists as well, since \(\mathbf{B}=\mathbf{T} \times \mathbf{N} .\) (a) Show that \(d \mathbf{B} / d s\) is perpendicular to \(\mathbf{B}(s)\) (b) Show that \(d \mathbf{B} / d s\) is perpendicular to \(\mathbf{T}(s) .\) [Hint: Use the fact that \(\mathbf{B}(s)\) is perpendicular to both \(\mathbf{T}(s)\) and \(\mathbf{N}(s)\) and differentiate \(\mathbf{B} \cdot \mathbf{T}\) with respect to \(s .1\) (c) Use the results in parts (a) and (b) to show that \(d\) B /ds is a scalar multiple of \(\mathbf{N}(s) .\) The negative of this scalar is called the torsion of \(\mathbf{r}(s)\) and is denoted by \(\tau(s) .\) Thus, $$\frac{d \mathbf{B}}{d s}=-\tau(s) \mathbf{N}(s)$$ (d) Show that \(\tau(s)=0\) for all \(s\) if the graph of \(\mathbf{r}(s)\) lies in a plane. INote: For reasons that we cannot discuss here, the torsion is related to the "twisting" properties of the curve, and \(\tau(s)\) is regarded as a numerical measure of the tendency for the curve to twist out of the osculating plane. \(]\)

Find parametric equations of the line tangent to the graph of \(\mathbf{r}(t)\) at the point where \(t=t_{0}\) $$\mathbf{r}(t)=e^{2 t} \mathbf{i}-2 \cos 3 t \mathbf{j}: t_{0}=0$$

Solve the vector initial-value problem for \(y(t)\) by integrating and using the initial conditions to find the constants of integration. $$y^{\prime}(t)=t^{2} \mathbf{i}+2 t \mathbf{j}, \quad \mathbf{y}(0)=\mathbf{i}+\mathbf{j}$$

The position vectors of two particles are given. Show that the particles move along the same path but the speed of the first is constant and the speed of the second is not. $$\begin{array}{ll} \mathbf{r}_{1}=2 \cos 3 t \mathbf{i}+2 \sin 3 t \mathbf{j} \\ \mathbf{r}_{2}=2 \cos \left(t^{2}\right) \mathbf{i}+2 \sin \left(t^{2}\right) \mathbf{j} & (t \geq 0) \end{array}$$
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