 Let's take a look at an example of how to use the trapezoidal rule to approximate the area under the curve of the function 1 divided by the quantity 16 plus x squared on the interval from 0 to 2. And we are going to use six subintervals to do that. Let's take a look at what we're trying to do by looking at the graph. So here is the graph of the function 1 divided by the quantity 16 plus x squared. We are doing this on the interval from 0 to 2 and since we are using six subintervals to approximate this area you can see the tick marks that are going to mark the beginning and end of each of our trapezoids. The first trapezoid then for instance will end right here almost rectangular and you'll recall that we can determine the height of each of these trapezoids with the little formula b minus a over n where n is the number of subintervals. So applying that here we obtain that each trapezoid is one third high which makes sense based upon how our graph is set up. The second trapezoid would end right here at two-thirds the third would end at one and so on and so forth. So let's go ahead and set up our rule that we're going to use. Recall that the trapezoidal rule only provides you with an approximation of the integral of the area under the curve therefore use of your approximately equal to symbol is necessary. Please don't forget that. If you think about how you find the area of a trapezoid that's really where the trapezoidal rule comes from itself. Area of a trapezoid is given to us by one half the height times the sum of the basis so that's why with the trapezoidal rule here in the calculus we have a half then we need to multiply by the height of each of those which we just determined to be one third. If you take a look at the graph again remember that height is really this horizontal distance you almost have to look at these with your head turned sideways to the left so the height is really that horizontal distance. Remember also and this is important in the trapezoidal rule the trapezoids are all of the same height. It is very possible you are going to come across problems in which you are simply asked to find a trapezoidal approximation and perhaps the data they provide you with yields trapezoids that are not the same height. That is very possible in the trapezoidal rule itself though the pure trapezoidal rule these trapezoids all have the same height please do remember that. As we proceed from there then remember that we need the length of the bases which is really these vertical heights that form in this case this would be the bottom base of the first trapezoid but the top base of the second here at two thirds this length would be the bottom base of the second trapezoid yet the top base of the third so you might remember that's why each of these in the middle here this one and this one all have to be doubled because they serve two purposes it's the bottom base of one trapezoid and the top base of another so it gets double counted for that reason. The bases that occur at x equals 0 and x equals 2 however simply count once. So the way in which we're going to write this up we need the function value at zero that's going to be the length of that very first base of the first trapezoid plus now we have to start multiplying by 2 and as we've seen we are going to evaluate this by using our graphing calculator and the built-in functions that it includes. So let's go ahead and switch to our graphing calculator and go ahead and do that. You'll need to have under y equals the function itself one divided by the quantity 16 plus x square please don't forget your parentheses around the denominator and then you can go to your quit screen and we'll go ahead and type out that which we had in the square brackets. Remember that we're using our built-in function notation so if you hit vars go across the top to y vars into function it's under y1 that we have our equation so we'll bring that up and our first one we need is y1 of 0. Remember that's telling the calculator that it needs to calculate the function value you have under y1 at 0. Now we just continue on from there but remember we need to do two times the next several ones so we have two times the function value now at one-third plus two times the function value at two-thirds plus two times the function value at one plus two times the function value at four-thirds plus two times the function value at five-thirds and finally the last one we only need once that's the end of the last trapezoid, the bottom base of that last trapezoid, at which we need the function value at 2. So if you hit Enter after you have that typed in, hopefully this is what you get. Now don't forget though we have to multiply by a half according to the trapezoidal rule, as well as the height of each of those individual trapezoids, which is going to be one-third. So in the end what we obtain for our approximation of this integral is approximately 0.116.