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Chapter 11: Problem 5

How do you find the indefinite integral of \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle ?\)

Short Answer

Expert verified
Now, suppose we have the following vector function: $$ \mathbf{r}(t) = \langle t^2, e^t, \sin(t)\rangle $$ Using the step-by-step solution, find the indefinite integral of the given vector function.

Step by step solution

01

Identify the Components of the Vector Function

Given the vector function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\), identify the components as \(f(t)\), \(g(t)\), and \(h(t)\).
02

Integrate Each Component Separately

Integrate each component with respect to the variable \(t\). This means finding the antiderivative \(F(t)\) for \(f(t)\), \(G(t)\) for \(g(t)\), and \(H(t)\) for \(h(t)\). These antiderivative functions will be used to construct the resulting vector function. $$ F(t) = \int f(t) dt \\ G(t) = \int g(t) dt \\ H(t) = \int h(t) dt $$
03

Construct the Resulting Vector Function

Combine the antiderivative functions \(F(t)\), \(G(t)\), and \(H(t)\) that were found in Step 2 to create a new vector function, which is the indefinite integral of the given function. Don't forget to include the constant of integration \(C\) for each component. $$ \mathbf{R}(t) = \langle F(t)+C_1, G(t)+C_2, H(t)+C_3\rangle $$
04

Result

The indefinite integral of the given vector function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) is: $$ \mathbf{R}(t) = \langle \int f(t) dt + C_1, \int g(t) dt + C_2, \int h(t) dt + C_3\rangle $$

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Most popular questions from this chapter

Relationship between \(\mathbf{r}\) and \(\mathbf{r}^{\prime}\) Consider the helix \(\mathbf{r}(t)=\langle\cos t, \sin t, t\rangle,\) for \(-\infty

Find the function \(\mathbf{r}\) that satisfies the given conditions. $$\mathbf{r}^{\prime}(t)=\left\langle 1,2 t, 3 t^{2}\right\rangle ; \mathbf{r}(1)=\langle 4,3,-5\rangle$$

Suppose the vector-valued function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) is smooth on an interval containing the point \(t_{0} .\) The line tangent to \(\mathbf{r}(t)\) at \(t=t_{0}\) is the line parallel to the tangent vector \(\mathbf{r}^{\prime}\left(t_{0}\right)\) that passes through \(\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right) .\) For each of the following functions, find an equation of the line tangent to the curve at \(t=t_{0} .\) Choose an orientation for the line that is the same as the direction of \(\mathbf{r}^{\prime}\). $$\mathbf{r}(t)=\left\langle 3 t-1,7 t+2, t^{2}\right\rangle ; t_{0}=1$$

Properties of dot products Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) \(\mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle,\) and \(\mathbf{w}=\left\langle w_{1}, w_{2}, w_{3}\right\rangle .\) Prove the following vector properties, where \(c\) is a scalar. $$\mathbf{u} \cdot \mathbf{v}=\mathbf{v} \cdot \mathbf{u}$$

Let \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\). a. Assume that \(\lim \mathbf{r}(t)=\mathbf{L}=\left\langle L_{1}, L_{2}, L_{3}\right\rangle,\) which means that \(\lim _{t \rightarrow a}|\mathbf{r}(t)-\mathbf{L}|=0 .\) Prove that \(\lim _{t \rightarrow a} f(t)=L_{1}, \quad \lim _{t \rightarrow a} g(t)=L_{2}, \quad\) and \(\quad \lim _{t \rightarrow a} h(t)=L_{3}\). b. Assume that \(\lim _{t \rightarrow a} f(t)=L_{1}, \lim _{t \rightarrow a} g(t)=L_{2},\) and \(\lim _{t \rightarrow a} h(t)=L_{3} .\) Prove that \(\lim _{t \rightarrow a} \mathbf{r}(t)=\mathbf{L}=\left\langle L_{1}, L_{2}, L_{3}\right\rangle\) which means that \(\lim _{t \rightarrow a}|\mathbf{r}(t)-\mathbf{L}|=0\).
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