Method: Riemann Sums

How you choose the evaluation points, $x_i$, is the only difference between all of the different Riemann Sum methods, left, right, middle, upper, or lower. This means the only step in your Riemann Sums Method that changes will be Step 5 when you choose the actual evaluation points, $x_i$.

You will notice that you do not actually need to know what your graph looks like in order to complete this process, and I generally don’t care what the actual graph is doing. You can find out all the data you need to successfully solve a Riemann Sum without ever graphing it. The only time I care what the actual graph looks like is if they ask me to actually draw the rectangles that go with the process. However, this is usually just a requirement your calculus teacher will ask on an in-class exam, and not usually something you are asked to do on the actual AP Calculus exam.

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 base from your $A=b\bullet h$ formula, using one of these formulas:

 $∆x=\frac{b–a}{\mathit{\#intervals}}$ OR $∆x=\frac{b–a}{\mathit{\#rectangles}}$ 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 rectangles 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 on the number line 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 rectangles that your problem calls for. If you are being directed to find your estimation using 6 rectangles, 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 base you used 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 rectangle , $A=b\bullet h=∆x\bullet f\left(x_i\right)$. If you were being asked to estimate based on 6 rectangles, then the list should look like this to start:

• $A_1=∆xf\left(\right)=$
• $A_2=∆xf\left(\right)=$
• $A_3=∆xf\left(\right)=$
• $A_4=∆xf\left(\right)=$
• $A_5=∆xf\left(\right)=$
• $A_6=∆xf\left(\right)=$

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

The only difference will be the evaluation point, $x_i$, 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 ( ONLY STEP THAT CARES WHAT TYPE of Riemann Sum ): Choose your evaluation points, $x_i$, based on the directions of your specific problem, either left, right, middle, upper, or lower.

• For a left-hand sum look at each subinterval and choose the left endpoint of that subinterval.
• For a right-hand sum look at each subinterval and choose the right endpoint of that subinterval.
• For a middle sum look at each subinterval and choose the midpointof that subinterval.

Add the left endpoint and the right endpoint of the subinterval and divide by 2 to find the midpoint value.

• For an upper sum look at each subinterval and choose the evaluation point, $x_i$,that would provide you the largest height , the largest $f\left(x_i\right)$, for that subinterval.
• For a lower sum look at each subinterval and choose the evaluation point, $x_i$,that would provide you the smallest height , the smallest $f\left(x_i\right)$, for that subinterval.

Plug those selected evaluation points, $x_i$, into your list of area formulas from Step 4 .

Step 6: Get your actual area value for each of your different area formulas by plugging your evaluation points, $x_i$, into the actual equation, $f\left(x\right)$, to find your height , and then multiply those values by your $∆x$, your base , to find the area, $A=b\bullet h=∆x\bullet f\left(x_i\right)$.

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

Step 8 (only if asked): Draw the rectangles for each subinterval that you just found the areas of.

• Start on the x-axis at the evaluation point, $x_i$, that you chose for each subinterval in Step 5 .
• Draw up or down from that point until you hit your actual graph, $f\left(x\right)$, then draw a horizontal line at that point that covers that rectangle’s subinterval, and then draw back to the x-axis . Repeat the process for each of your subintervals.

I always say, “Draw up. Draw over. Draw down.”