   Chapter 12.2, Problem 64E

Chapter
Section
Textbook Problem

# Integration How do you find the integral of a vector-valued function?

To determine

The value of the integral of a vector-valued function.

Explanation

Let r(t)=f(t)i+g(t)j+h(t)k be the vector-valued function.

If f(t), g(t) and h(t) are continuous on the closed interval [a,b] where atb, then the indefinite integral of the function r(t) is calculated component-by-component as,

r(t)dt=[f(t)dt]i+[g(t)dt]j+[h(t)dt]k=[F(t)+C1]i+[G(t)+C2]j+[H(t)+C3]k=F(t)i+G(t)j+H(t)k+C1i+C2j+C3k=F(t)i+G(t)j+H(t)k+C

Where F(t)=f(t)dt, G(t)=g(t)dt and H(t)=h(t)dt and C=C1i+C2j+C3k is an arbitrary constant vector.

For example,

Consider the indefinite integral:

r(t)dt=(3t2i+j2tk)dt

Integrate the above indefinite integral component-by-component to get:

(3t2i+j2tk)dt=[3t2dt]i+[dt]j+[(2t)dt]k=3(t33)i+tj2(t22)k+(C1i+C2j+C3k)=t3i+tjt2k+C

Where,C is the constant of integration

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started 