5.2 Definite Integrals

5.2 Definite Integrals

When we find the area under a curve by adding rectangles, the answer is called a Riemann sum. The width of a rectangle is called a subinterval. Subintervals are often denoted by xbecause they represent the change in x…but you all know this at least from chemistry class, right? subinterval interval Now let’s do some more notation so that you will understand it when you see it in the text book…

rectangle base Summation of n rectangles rectangle height When we find the area under a curve by adding rectangles, the answer is called a Riemann sum. The width of a rectangle is called a subinterval. If we take n rectangles (subintervals) and add them all up, the summation would look like this: subinterval interval

When we find the area under a curve by adding rectangles, the answer is called a Riemann sum. The width of a rectangle is called a subinterval. Subintervals do not all have to be the same size. subinterval Equal subintervals make for easier and faster calculation, but some curves call for rectangles of different bases depending upon the shape of the curve. While we won’t be doing any in this lesson, we should at least consider that possibility here. interval In calculus texts, the partition xis also denoted by P. Let’s look at one summation expression that uses this notation…

If the partition is denoted by 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. We can also think of the size of this partition in the same way as we think of x And now for some more terminology: subinterval interval if P is a partition of the interval

if P is a partition of the interval Do you remember how we can improve this approximation? Increase the number of subintervals subinterval subinterval interval interval

if P is a partition of the interval As n gets bigger, P gets smaller Since we know how to take limits… DON’T WE? We can then send the number of partitions to infinity which will send the base of each rectangle (the size of each partition) to… subinterval interval 0

is called the Riemann Sum of over . If we use subintervals of equal length, then the length of a subinterval is: The sum can then be given by:

Leibnitz introduced a simpler notation for the definite integral: The Definite Integral over the interval [a,b] Notice how as x 0, the change in xbecomes dx.

So what do all of these symbols mean? upper limit of integration Integration Symbol integrand variable of integration lower limit of integration

Rectangle Height Summation Symbol Rectangle Base Don’t forget that we are still finding the area under the curve. For #8-28 in 5.2, if you make a graph of the problem, you can find the integral easily.

This happens to be #1 in 5.2 but we will do it to help you through the notation when you do the homework: over the interval [0, 2] Rectangle Height Rectangle Base …which means just find the area under the curve y = x2from 0 to 2

5.2 Definite Integrals