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17Calculus - Integrals of Vector Functions

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Coordinate Systems

Vectors

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Double Integrals - 2Int

Triple Integrals - 3Int

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On this page we explain how to evaluate integrals of vector functions.

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.

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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

\( \hat{i} + \hat{j} + \hat{k} \)

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\(\displaystyle{ \int{ \frac{5}{t^2}\vhat{i} - 4\sqrt{t}\vhat{j} ~dt } }\)

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

\(\displaystyle{ \left[ \frac{-5}{t}+c_1\right] \vhat{i} - \left[ \frac{8}{3}t^{3/2} + c_2\right] \vhat{j} }\)

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\(\displaystyle{ \int{ \frac{2}{t}\vhat{i} - \sin(t)\vhat{j} + \sec^2(2t)\vhat{k} ~dt } }\)

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

\(\displaystyle{ \left[ 2\ln(t)+c_1\right]\vhat{i} + \left[ \cos(t)+c_2 \right]\vhat{j} + \left[ (1/2)\tan(2t) + c_3\right]\vhat{k} }\)

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Practice Instructions

Unless otherwise instructed, evaluate these integrals.

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