AP®MathsCollege BoardCalculus BCStudy GuidesUnit 9: Parametric Equations, Vector-Valued Functions & Polar CoordinatesVector-Valued FunctionsIntegrating Vector-Valued Functions

Integrating Vector-Valued Functions (College Board AP® Calculus BC): Study Guide

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on

Integrating vector-valued functions

How do I integrate vector-valued functions?

  • To integrate a vector-valued function, integrate both components separately

integral open angle brackets x open parentheses t close parentheses comma space y open parentheses t close parentheses close angle brackets space d t equals open angle brackets integral x open parentheses t close parentheses space d t comma space integral y open parentheses t close parentheses space d t close angle brackets

  • You must remember to add a constant of integration to both components

Examiner Tips and Tricks

When integrating vector-valued functions, it helps to use different letters for the different constants of integration (plus C, plus D, ... etc).

How do I find the constants of integration?

  • To find the constants of integration, in the question you will be given a known vector value for a particular value of t

    • e.g. the vector-valued function is open angle brackets 6 comma space 0 close angle brackets when t equals 2

    • Use this information to form and solve equations to find the two constants

Worked Example

The derivative of a vector-valued function is open angle brackets 12 t cubed minus 2 t comma space 1 plus 3 square root of t close angle brackets.

At t equals 1, the function is open angle brackets negative 3 comma space 7 close angle brackets.

Find the function.

You are given the derivative of a vector-valued function, so you need to integrate it to find the original vector-valued function

Integrate both components (write square root of t as t to the power of 1 half end exponent) and add a different constant of integration to each component

open angle brackets 3 t to the power of 4 minus t squared plus C space comma space t plus 2 t to the power of 3 over 2 end exponent plus D close angle brackets

You are given a known vector value, open angle brackets negative 3 comma space 7 close angle brackets, when t equals 1

First substitute in t equals 1

open angle brackets 3 open parentheses 1 close parentheses to the power of 4 minus open parentheses 1 close parentheses squared plus C space comma space open parentheses 1 close parentheses plus 2 open parentheses 1 close parentheses to the power of 3 over 2 end exponent plus D close angle brackets equals open angle brackets 2 plus C space comma space 3 plus D close angle brackets

Then set this equal to open angle brackets negative 3 comma space 7 close angle brackets

open angle brackets 2 plus C space comma space 3 plus D close angle brackets equals open angle brackets negative 3 space comma space 7 close angle brackets

Solve the two equations to find C and D

table row cell 2 plus C end cell equals cell negative 3 rightwards double arrow C equals negative 5 end cell row cell 3 plus D end cell equals cell 7 rightwards double arrow D equals 4 end cell end table

Substitute the values of C and D back into the function to get the final answer, which you should write in vector notation

open angle brackets 3 t to the power of 4 minus t squared minus 5 space comma space t plus 2 t to the power of 3 over 2 end exponent plus 4 close angle brackets

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    Author
    Reviewer
    Mark Curtis

    Author: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.