Properties of Definite Integrals (College Board AP® Calculus BC): Revision Note

Definite integrals

What is a definite integral?

• A definite integral is written in the form

  • ${\int }_a^{b}f\left(x\right)dx$

    • $f\left(x\right)$ is the function being integrated (also known as the integrand)

    • $dx$ indicates that the function is being integrated with respect to $x$

    • $a$ and $b$are the integration limits

      • the function is integrated 'from $a$ to $b$'

    • You'll most often see definite integrals with $a\le b$

      • But integrals with $a>b$ are also valid

• A definite integral can be interpreted in a number of ways

  • It is a mathematical operation that outputs a number

    • based on $f\left(x\right)$ and the values of $a$ and $b$

  • If $f\left(x\right)$ is interpreted as a rate of change function

    • then the definite integral is the accumulation of change as $x$ goes from $a$ to $b$

  • On the graph of $y=f\left(x\right)$, where $f\left(x\right)\ge 0$ on the interval $\left[a,b\right]$

    • the definite integral gives the area between $y=f\left(x\right)$and the $x$-axis, between $x=a$ and $x=b$

How is a definite integral defined as a limit of Riemann sums?

• The value of a definite integral can be defined as a limit of Riemann sums:

  • ${\int }_a^{b}f\left(x\right)dx=\underset{\max \mathrm{\Delta }{x}_i\to 0}{\lim }\sum _{i=1}^{n}f\mathit{\left(}{x}_i^{\mathit{\ast }}\mathit{\right)}\mathrm{\Delta }{x}_i$

    • $n$ is the number of subintervals used for the sum

      • This is the same as the number of rectangles

    • $x_i^{\mathit{\ast }}$ is a value of $x$ in the $i$th subinterval

      • note that this is any value in the $i$th subinterval

    • $\mathrm{\Delta }{x}_i$ is the width of the $i$th subinterval

      • i.e. the width of the $i$th rectangle

    • $\underset{\max \mathrm{\Delta }{x}_i\to 0}{\lim }$ means the limit as the width of the largest subinterval (and hence of all subintervals) approaches zero

  • As the subintervals (and rectangles) get narrower and narrower

    • the area of the approximating rectangles gets closer and closer to the exact area under the curve

• Note that having the width of the largest subinterval approach zero

  • is equivalent to having the number of subintervals approach infinity

• Therefore the Riemann sum limit for a definite integral can also be written with $n$ as the limit index

  • ${\int }_a^{b}f\left(x\right)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^{n}f\mathit{\left(}{x}_i^{\mathit{\ast }}\mathit{\right)}\mathrm{\Delta }{x}_i$

    • If the subintervals are assumed to be of equal width, then expressions for $f\mathit{\left(}{x}_i^{\mathit{\ast }}\mathit{\right)}$ and $\mathrm{\Delta }{x}_i$ can be written in terms of $a$, $b$ and $n$

    • See the Worked Example

Definite Integrals of sums, differences and constant multiples

What are the properties of sums, differences and constant multiples of definite integrals?

• These are related to the equivalent properties for indefinite integrals

Definite integral of a constant times a function

• If $k$ is a constant, then

  • ${\int }_a^{b}kf\left(x\right)dx=k{\int }_a^{b}f\left(x\right)dx$

    • I.e. the constant can be brought out in front of the integral as a multiplier

Definite integral of a sum or difference of functions

• If $f$ and $g$ are two functions being integrated over the same interval, then

  • ${\int }_a^b\left(f\left(x\right)\pm g\left(x\right)\right)dx={\int }_a^{b}f\left(x\right)dx\pm {\int }_a^{b}g\left(x\right)dx$

    • I.e. the integral of a sum (or difference) is equal to the sum (or difference) of integrals

Changing limits of integration

What are the definite integral properties involving integration limits?

• There are three properties you should know here

Definite integral of a zero-length interval

• For any function $f$

  • ${\int }_a^{a}f\left(x\right)dx=0$

    • I.e. if the top and bottom integration limits are equal, the definite integral is equal to zero

Reversing the limits of integration

• For a function $f$

  • ${\int }_b^{a}f\left(x\right)dx=-{\int }_a^{b}f\left(x\right)dx$

    • I.e. reversing the integration limits 'flips' the sign of the value of the definite integral

Definite integrals on adjacent intervals

• For a function $f$, and for $c$ such that $a\le c\le b$

  • ${\int }_a^{b}f\left(x\right)dx={\int }_a^{c}f\left(x\right)dx+{\int }_c^{b}f\left(x\right)dx$

    • I.e. the sum of definite integrals over adjacent subintervals of

      $\left[a,b\right]$ is equal to the total definite integral from $a$ to $b$