t dt Estimate the value of the integral with the trapezoidal rule. Use n = 4.

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1 Trapezoidal Rule We have already found the value of an integral using rectangles in the first lesson of this module. In this section we will again be estimating the value of an integral using geometric formulas but this time we will use trapezoids. Trapezoids give a better estimate of the area under a function when the function is exponential because the shape of a trapezoid fits the shape of the curve better. Simpson's Rule is another way of doing numerical differentiation and is covered at the end of the lesson in the book, but it is not on the AB or BC syllabi, so we will not be covering it here. The basic format of this lesson is the same as in lesson 5-. I will go through the process of using the trapezoidal rule and then give you the shortcut formula that can be derived, and then do some homework examples. Throughout this lesson, you are to only use the trapezoidal rule - do not use Simpson's Rule.

2 A: ( t + ) t dt Estimate the value of the integral with the trapezoidal rule. Use n = 4. Graph the function y = t + t : / / As you can see I have sub-divided the interval from to into 4 sub-intervals (the problem will always tell you how many intervals to use). The trapezoid formed by each sub-interval is outlined in red. I want to see your trapezoids on your paper! The general formula for the area of a trapezoid is A= hb ( + b), where h is the height of the trapezoid (the distance between the two bases) and b and b are the lengths of the two parallel bases (vertical lines at the beginning and end of our sub-interval). In our graph we can see that the height of each trapezoid will be the length of the sub-interval (/ in this case) and that the bases will be found by finding the value of f (t) at the beginning and end of each sub-interval. ** The trapezoids are lying on their sides; the bases are the vertical lines. ** So, the total area will be: Area of Trapezoid = / * / * (f() + f(/)) = /4 ( + 5/8) Area of Trapezoid = / * / * (f(/) + f()) = /4 ( 5/8 + ) Area of Trapezoid = / * / * (f() + f(/)) = /4 ( + 9/8) Area of Trapezoid 4 = / * / * (f(/) + f()) = /4 (9/8 + ) The total area will be the sum of the above areas. I am going to factor out the /4 because it will make the addition easier. Total area = /4 ( + 5/8 + 5/ /8 + 9/8 + ) = h ( + 4) = ** You do not have to do the arithmetic! Use your calculator to get a decimal answer if you want to! Just remember to give the answer correct to three places.

3 Notice that in the first chart the second base of one trapezoid is the same as the first base of the next trapezoid. That is the basis for a handy formula. A shortcut yea!! You can factor out the (/) h from each trapezoid and you will then have the sum of all the bases, and there will be two of each of the inside bases. Looking at it that way, our problem would have looked like this: A= f() + f + f() + f + f() A = + + () + + = That method saves some time; just remember that only the inside bases are doubled. The first and last ones are not. In general, the formula looks like: Trapezoidal Rule A= b( f( a) + f( b) f( y) + f( z) ) Where [a, z] is our interval and the subintervals are at a, b, c,. y, z That's all there is to it. If you can do basic geometry, you can do this. The trick is to make sure that you keep track of what you are doing.

4 LRAM, RRAM, and the Trapezoidal Rule There is one more relationship that is nice to know, one between the LRAM, RRAM, and the trapezoidal rule. Estimate with the Trapezoidal Rule LRAM + RRAM T = When all estimates use the same number of subintervals. The Trapezoidal rule gives you an estimate that is the average of the upper and lower sums that we got with rectangles. Pretty slick! Note #: Wow read that last paragraph again! I just said that the trapezoidal rule will give us the average of the upper and lower sums. How can that be? We have already learned that a lower sum is not always found with LRAM; it could be given by either LRAM or RRAM. The same is true of upper sums and RRAM. So how I am getting away with saying this? If you know, send me the answer with the heading Trap Rule for a point of EC. Hint: You will need this relationship for one of the quiz questions

5 Homework Examples #, 7 #: (a) Use the Trapezoidal Rule with n = 4 to approximate the value of the integral, (b) use the concavity of the function to predict whether the approximation is an overestimate or underestimate, and (c) find the integral's exact value to check the answers. x dx This is our function. Make a quick sketch to see what you are doing. Then draw in the trapezoids. For n = 4, our separations will be at x =, ½,, /, The height = ½ ( ) = f x x A= f + f + f + f + f ( ) ( ) ( ) 7 A = + + ( ) = The Trapezoidal rule is an overestimation. 4 x dx = x = 4 = 4 4 Find the approximation with the Trapezoidal Rule. Note: on the AP exam, you can leave your answer at this first step as long as you have defined f (x) somewhere in a step above! Since the curve is concave up, we can see that the trapezoids extend over the top of the curve and include too much area. Integrate by finding the antiderivative and evaluating it. This confirms our answer to part (b). The exact answer is less than the trapezoidal estimate.

6 #7: A rectangular swimming pool is ft wide and 5 ft long. The table below shows the depth of the water at 5-ft intervals from one end of the pool to another. Estimate the volume of water in the pool using the Trapezoidal Rule with n =, applied to the integral V = h( x) dx 5 5 V = h( x) dx Height = (8.) + (9.) + (9.9 + (.5) + () V = 5 + (.5) + (.9) + (.) + (.7) + V 5,99 ft ( ) ( ) This is our integral. I am going to move the constant out in front. Then we will use the Trapezoidal Rule on the integral of h (x) and multiply that by to get the volume. The values of h (x) are in the chart in the book, as are the divisions every 5 feet. Again, on the AP exam you are allowed to leave the answer at the first step unless they have specifically asked you to give the answer correct to the nearest foot (which would require a decimal answer).