5.2 Definite Integrals Created by Greg Kelly, Hanford High School, Richland, Washington Revised by Terry Luskin, Dover-Sherborn HS, Dover, Massachusetts

When we find the area under a curve by adding rectangles, the answer is called a Riemann sum. subinterval partition The width of a rectangle is called a subinterval. The width of the entire interval is called the partition. Subintervals do NOT all have to be the same size.

subinterval partition If the partition is symbolized P, then the length of the longest subinterval is called the norm of P and is denoted by. As gets smaller, the approximation for the area gets better!

If we use subintervals of equal length, then the length of a subinterval is: The definite integral of f over the interval [a, b] is: We can move through each subinterval by adding a subscript k to the horizontal coordinate c, and then let k vary from 1 through the last subinterval n. Eventually, we’d let n increase without bound, approaching infinity!

Leibniz introduced a simpler notation for the definite integral: Note that the very small change in x becomes dx.

Integration Symbol lower limit of integration upper limit of integration integrand variable of integration (dummy variable) This is called a dummy variable because the result does not depend on the variable chosen.

We have the notation for integration, but we still need to learn how to evaluate the integral.

time velocity After 4 seconds, the object has gone 12 feet. In section 5.1, we considered an object moving at a constant rate of 3 ft/sec. Since rate. time = distance: If we draw a graph of the velocity, the distance that the object travels is equal to the area under the velocity “curve”.

If the velocity varies linearly: Distance traveled: ( C=0 since s=0 at t=0 ) After 4 seconds: The net distance is still equal to the area under the “curve”! s =.5t 2 /2 +1t + 0 s = 0.25t 2 + t + 0

What if: Could we find the area under this velocity curve from 0 to 4?

The area under the curve We can use anti-derivatives to find the area under a derivative curve! s = 20/3

Let’s look at it another way: Then: Let equal the area under the curve from a to x.

This is the definition of derivative! Take the anti-derivative of both sides to find an explicit formula for area. Net area under f(x) from a to b = antiderivative minus antiderivative of f(x) at b of f(x) at b

Net Area

Area from x=0 to x=1 Example: Find the area under the curve from x = 1 to x = 2. Area from x=0 to x=2 Area under the curve from x = 1 to x = 2.

Example: Find the area under the curve from x = 1 to x = 2. To use the TI-83/84: fnInt(x 2,x,1,2) =

Example: Find the total area between the x-axis and the curve from to. On the TI-83/84: If you use the absolute value function, you don’t need to reverse the sign of the integral from π/2 to 3π/2. pos. neg. 

min f max f The area of a rectangle drawn under the curve (would be less than the actual area under the curve. The area of a rectangle drawn above the curve would be more than the actual area under the curve. h