17Calculus - Integrals of Vector Functions
17Calculus
On this page we explain how to evaluate integrals of vector functions.
Recommended Books on Amazon (affiliate links) | ||
|---|---|---|
 |
 |
 |
Integrals of vector functions use special techniques. Here is a video clip that should help you a lot.
Dr Chris Tisdell - integrals of vector functions [6mins-57secs]
video by Dr Chris Tisdell |
|---|
To put this all together, here is a full lecture on derivatives and integrals of vector functions.
Prof Leonard - Calculus 3 Lecture 12.2: Derivatives and Integrals of Vector Functions [2hrs-42mins-18secs]
video by Prof Leonard |
|---|
Try these practice problems.
Practice
Unless otherwise instructed, evaluate these integrals.
\(\displaystyle{ \int_{0}^{\pi/2}{ \vec{r}(t)~dt } }\) for \( \vec{r}(t) = [3\sin^2t\cos t]\,\hat{i}+ \) \( [3\sin t\cos^2t]\,\hat{j} + \) \( [2\sin t\cos t]\,\hat{k} \)
Problem Statement |
|---|
Evaluate \(\displaystyle{ \int_{0}^{\pi/2}{ \vec{r}(t)~dt } }\) for \( \vec{r}(t) = [3\sin^2t\cos t]\,\hat{i}+ \) \( [3\sin t\cos^2t]\,\hat{j} + \) \( [2\sin t\cos t]\,\hat{k} \)
Final Answer |
|---|
\( \hat{i} + \hat{j} + \hat{k} \)
Problem Statement
Evaluate \(\displaystyle{ \int_{0}^{\pi/2}{ \vec{r}(t)~dt } }\) for \( \vec{r}(t) = [3\sin^2t\cos t]\,\hat{i}+ \) \( [3\sin t\cos^2t]\,\hat{j} + \) \( [2\sin t\cos t]\,\hat{k} \)
Solution
Krista King Math - 2026 video solution
video by Krista King Math |
|---|
Final Answer
\( \hat{i} + \hat{j} + \hat{k} \)
Log in to rate this practice problem and to see it's current rating. |
|---|
\(\displaystyle{ \int{ \frac{5}{t^2}\vhat{i} - 4\sqrt{t}\vhat{j} ~dt } }\)
Problem Statement |
|---|
Evaluate \(\displaystyle{ \int{ \frac{5}{t^2}\vhat{i} - 4\sqrt{t}\vhat{j} ~dt } }\)
Final Answer |
|---|
\(\displaystyle{ \left[ \frac{-5}{t}+c_1\right] \vhat{i} - \left[ \frac{8}{3}t^{3/2} + c_2\right] \vhat{j} }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \frac{5}{t^2}\vhat{i} - 4\sqrt{t}\vhat{j} ~dt } }\)
Solution
MIP4U - 2037 video solution
video by MIP4U |
|---|
Final Answer
\(\displaystyle{ \left[ \frac{-5}{t}+c_1\right] \vhat{i} - \left[ \frac{8}{3}t^{3/2} + c_2\right] \vhat{j} }\)
Log in to rate this practice problem and to see it's current rating. |
|---|
\(\displaystyle{ \int{ \frac{2}{t}\vhat{i} - \sin(t)\vhat{j} + \sec^2(2t)\vhat{k} ~dt } }\)
Problem Statement |
|---|
Evaluate \(\displaystyle{ \int{ \frac{2}{t}\vhat{i} - \sin(t)\vhat{j} + \sec^2(2t)\vhat{k} ~dt } }\)
Final Answer |
|---|
\(\displaystyle{ \left[ 2\ln(t)+c_1\right]\vhat{i} + \left[ \cos(t)+c_2 \right]\vhat{j} + \left[ (1/2)\tan(2t) + c_3\right]\vhat{k} }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \frac{2}{t}\vhat{i} - \sin(t)\vhat{j} + \sec^2(2t)\vhat{k} ~dt } }\)
Solution
MIP4U - 2038 video solution
video by MIP4U |
|---|
Final Answer
\(\displaystyle{ \left[ 2\ln(t)+c_1\right]\vhat{i} + \left[ \cos(t)+c_2 \right]\vhat{j} + \left[ (1/2)\tan(2t) + c_3\right]\vhat{k} }\)
Log in to rate this practice problem and to see it's current rating. |
|---|
Really UNDERSTAND Calculus
Topics You Need To Understand For This Page
To bookmark this page and practice problems, log in to your account or set up a free account.
Search Practice Problems
Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
|
| |
free ideas to save on books |
|---|
|
|---|
|
|---|
Practice Instructions
Unless otherwise instructed, evaluate these integrals.
