definite integral

A definite integral of rate of change gives the net change in from to . It also expresses the area under the curve of the function

Interpretation

Example

At time , a population of bacteria grows at the rate of  grams per day, where  is measured in days.

• is the accumulation of weight change of bacteria (unit: grams) over time, from day 0 to day 8
  • is the area under the curve of the graph for , from to

Riemann sum

Overview

An approximation of the area under a curve by dividing it into simple shapes, and summing those composite areas .

where , the width of each shape, usually rectangles.

From definite integral to Riemann sum

From Riemann sum to definite integral

Examples

Detailed example

We can convert between definite integral and Riemann sum (where is the number of rectangles):

Turn the original sum () to this form (), by changing the sign of :

From this equation, we can identify the number of rectangles for the Riemann sum is . Similarly, we extract the width of each rectangle:

Using the width of each rectangle to find endpoints:

We also extract the height of each rectangle:

Rewrite the function in terms of and , using the equation :

From here, we can deduce that and . Thus . We define the definite integral as:

Link to original

Properties

(Khan Academy)

Sum/Difference

Constant multiple

Reverse interval

Zero-length interval

Add intervals

Notable Functions

Piecewise functions

(Khan Academy)

Example

x+1 & \text {for} & x<0 \ \cos(\pi x) & \text {for} & x\geq 0 \end{cases}$$

\int_{-1}^{1}f(x)dx &= \int_{-1}^{0}(x+1)dx+\int_{0}^{1}\cos(\pi x)dx\ &=\frac{(x+1)^{2}}{2}\bigg|{-1}^{0}+\frac{\sin(\pi x)}{\pi}\bigg|{0}^{1}\ &=\frac{1}{2}+\frac{\sin(\pi)}{\pi}-\frac{\sin 0}{\pi}\ &=\frac{1}{2} \end{align*}$$

Application

Accumulation problem with definite integral

(Khan Academy) Given is the rate of change of , according to FTC Part 2, we can calculate at a given point of time using initial condition and definite integral:

Example 1

The population of a town grows at a rate of  people per year (where  is time in years). At time , the town’s population is 1200. What is the town’s population at ?

Example 2

The depth of the water in a tank is changing at a rate of  cm/min (where  is the time in minutes). At time , the depth of the water is 35 cm. What is the change in the water’s depth during the 4th minute?

Change in the water’s depth during the 4th minute is given by:

Find the area under the curve with definite integral

The area under the curve of between and (with ) is