Method: Trapezoidal Riemann Sums

The Riemann Sum method for using trapezoids is almost identical to the method used for using rectangles . The only real difference will be on Step 4 when you set up the area formula , and Step 5 when you choose your evaluation points .

Step 1: Draw a number line with the left and right endpoints of the number line coming from your given interval, [ a , b ].

Step 2: Most of the time you will need to calculate your $∆x$, the height from your $A_n=\frac{1}{2}\left(\mathit{base}\mathrm{ }1+\mathit{base}\mathrm{ }2\right)\bullet \mathit{height}$ formula, using one of these formulas: $∆x=\frac{b–a}{\mathrm{\#}\mathit{intervals}}$ OR $∆x=\frac{b–a}{\mathrm{\#}\mathit{trapezoids}}$ OR $∆x=\frac{b–a}{n}$.

They are all the same formula, but the language that is used in your problem will vary.

The n-value represents the number of trapezoids or subintervals you are being asked to use.

• Sometimes you are just given a table of values to work from. In that situation you will mark subintervals based on the x-value subintervals provided by the table, and you will not calculate a standard $∆x$.

Step 3: Mark your subintervals on your number line.

• If you have a $∆x$ .

Always start at the left endpoint of your number line, move down your number line making a mark on the number line every $∆x$, until you reach the right endpoint of your number line. If you have done everything properly you should make your final mark exactly on the rightendpoint of your number line.

Label the gaps between each mark $A_1, A_2, A_3,...A_n$. If you have done everything properly the last gap should have the same subscript number as the number of intervals or trapezoids that your problem calls for. If you are being directed to find your estimation using 6 trapezoids, then your last gap should be labeled $A_6$.

• If you have a table of values, you will want to label a number line based on the independent variable values ( x-values ) of that table. When using a table of values, you might see inconsistent gaps between those values, and in that situation, you will have to adjust the height you use on each of your subarea rectangles .

Label the gaps on the number line between each mark $A_1, A_2, A_3,...A_n$.

Step 4: Start laying out your area formulas for each of your subintervals.

I call this The Assembly Line Process. You will be writing the same things over and over again because all of the areas you are calculating are using the same area formula for a trapezoid , $A=\frac{1}{2}\left(\mathit{base }1+\mathit{base }2\right)\bullet \mathit{height}=\frac{1}{2}\left(f\left(x_i\right)+f\left(x_{i+1}\right)\right)\bullet ∆x$. If you were being asked to estimate based on 6 trapezoids , then the list should look like this to start:

• $A_1=\frac{1}{2}\left(f\left(\right)+f\left(\right)\right)\bullet ∆x=$
• $A_2=\frac{1}{2}\left(f\left(\right)+f\left(\right)\right)\bullet ∆x=$
• $A_3=\frac{1}{2}\left(f\left(\right)+f\left(\right)\right)\bullet ∆x=$
• $A_4=\frac{1}{2}\left(f\left(\right)+f\left(\right)\right(\bullet ∆x=$
• $A_5=\frac{1}{2}\left(f\left(\right)+f\left(\right)\right)\bullet ∆x=$
• $A_6=\frac{1}{2}\left(f\left(\right)+f\left(\right)\right)\bullet ∆x=$

Notice that all of these trapezoid area formulas look exactly the same.

The only difference will be the evaluation points, $x_i \mathrm{and }{x}_{i+1}$, that you choose to plug into your $f\left(\right)$.

I leave the evaluation points empty at this step and fill them in as a part of Step 5 .

Step 5: Choose your two evaluation points, $x_i \mathrm{and }{x}_{i+1}$, for each of your subintervals, and plug those values into your equations form Step 4.

The evaluation points for the trapezoid method will always be the left and right endpoints of each subinterval.

Step 6: Get your actual area value for each of your different area formulas by plugging your evaluation points, $x_i \mathrm{and }{x}_{i+1}$, into the actual equation, $f\left(x\right)$, to find your base 1 and base 2 values, then multiply those values by your $∆x$, your height , and finally multiply that by $\frac{1}{2}$ to find the areas,
$A_n=\frac{1}{2}\left(f\left(x_i\right)+f\left(x_{i+1}\right)\right)\bullet ∆x$.

Step 7: Add up the areas of all of your subinterval trapezoids to get an estimate for the total area between the curve and the x-axis .

Step 8 (only if asked): Draw the trapezoids that you just found the areas of. Start on the x-axis at the evaluation point , $x_i$, that you found in Step 5. Draw up from that point until you hit your actual graph, $f\left(x_i\right)$, then draw a line connecting that point with the other side of the trapezoid , your $f\left(x_{i+1}\right)$, that covers that trapezoid’s subinterval, and then draw back down to the x-axis . Repeat the process for each of your subintervals. I always say, “Draw up. Draw over. Draw down.”