Popular applications of integration
1) Area under a curve: Let's take the function f(x)=x2 (a parabola openning up from the origin). Let's say we wanted to find the area under this curve and above the x-axis from x=1 to x=3. To get this we set up the definite integral of x2 from x=1 to x=3. It will look like this:
∫13f(x)dx
To evaluate this definite integral we first integrate x2. Get the result (without the constant of integration), then substitute the upper limit of 3 into it and the lower limit of 1 into it and subtract them. It looks like this:
∫13(x2)dx
=x3/3. Now, substitute the limits into this expression and subtract them:(33/3)-(13/3)=9-1/3=27/3-1/3=26/3 square units of area. The TI-83 calculator will complete this in a matter of seconds and give you a graph of the area in question.
2)Finding total distance traveled by a particle given its velocity: Let's say a particle travels according the the velocity V=t2-4t-1. The total distance traveled will be the amount of area between the horizontal axis (t-axis in this case) and the velocity curve. Let's find the total distance traveled between t=2 and t=4.
We need to set-up the definite integral from 2 to 4 of the velocity function. It looks like this:
∫24(t2-4t-1)dt
=(t3/3-4t2/2-t). Now, substitute the limits into this expression and subtract, we get
[(43/3-4(42)/2-4]-[13/3-4(1)2