Example 1: Upper and Lower Riemann Sums

Since the only part of the Riemann Sums Method that changes from type to type is Step 5 . I will use a single equation to show you how Step 5 varies for each type of Riemann Sum.

Use 6 subintervals to estimate the area between the curve and the x-axis for

f ( x ) = ( x 3 ) 3 + 2 ( x 3 ) 2 + 1 on the interval [1,3]. Do this as an upper sum and lower sum.

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

 

The number line will be the same for all of the methods. The interval in these problems is [1,3].

 

 

 

 

Step 2: Calculate your x , the base from your A = b h formula, using the formula:

  x = b a #intervals

 

The number of intervals would be 6 since the problem asks us to use 6 subintervals.

 

x = b a #intervals

 

x = 3 1 6

 

x = 2 6

 

x = 1 3

Step 3: Mark your subintervals on your number line.

 

Here you have x = 1 3 . Starting at the left endpoint x = 1 and ending at the right endpoint x = 3 .

 

Label these gaps between each mark A 1 ,   A 2 ,   A 3 , A 4 ,   A 5 , A 6 .

 

 

 

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

A 1 = x f (   ) =

A 2 = x f (   ) =

A 3 = x f (   ) =

A 4 = x f (   ) =

A 5 = x f (   ) =

A 6 = x f (   ) =

 

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 an upper sum look at each subinterval and choose the evaluation point, x i ,that would provide you the largest height , the largest f ( x i ) , for that subinterval.

x 1 = 4 3

x 2 = 5 3

x 3 = 5 3

x 4 = 2

x 5 = 7 3

x 6 = 8 3

 

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 ( x i ) , for that subinterval.

x 1 = 1

x 2 = 4 3

x 3 = 2

x 4 = 7 3

x 5 = 8 3

x 6 = 3

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 ( x ) , to find your height , and then multiply those values by your x , your base , to find the area, A = b h = x f ( x i ) .

 

You use the same x = 1 3 as the base for each of the 5 options, and we also use the same equation, f ( x ) = ( x 3 ) 3 + 2 ( x 3 ) 2 + 1 , to find the heights .

A 1 = x f ( 4 3 ) = 1 3 52 27 = 52 81

A 2 = x f ( 5 3 ) = 1 3 59 27 = 59 81

A 3 = x f ( 5 3 ) = 1 3 59 27 = 59 81

A 4 = x f ( 2 ) = 1 3 2 = 2 3

A 5 = x f ( 7 3 ) = 1 3 43 27 = 43 81

A 6 = x f ( 8 3 ) = 1 3 32 27 = 32 81

A 1 = x f ( 1 ) = 1 3 1 = 1 3

A 2 = x f ( 4 3 ) = 1 3 52 27 = 52 81

A 3 = x f ( 2 ) = 1 3 2 = 2 3

A 4 = x f ( 7 3 ) = 1 3 43 27 = 43 81

A 5 = x f ( 8 3 ) = 1 3 32 27 = 32 81

A 6 = x f ( 3 ) = 1 3 1 = 1 3

 

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 .

 

A 1 = 52 81

A 2 = 59 81

A 3 = 59 81

A 4 = 2 3

A 5 = 43 81

A 6 = 32 81

A Total 52 81 + 59 81 + 59 81 + 2 3 + 43 81 + 32 81 299 81

 

A 1 = 1 3

A 2 = 52 81

A 3 = 2 3

A 4 = 43 81

A 5 = 32 81

A 6 = 1 3

A Total 1 3 + 52 81 + 2 3 + 43 81 + 32 81 + 1 3 235 81

 

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 ( x ) , 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.”

 

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