Area Under the Curve

Area Under the Curve 1
AREA UNDER THE CURVE
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Area Under the Curve 2
Area under the Curve
Approximate area when n1 = 12
When n = 12, it means that there are 12 rectangular strips between x = 0 and x = 6. Since the
rectangular strips are of the equal width, the width of each strip is
( 6−0 ) units
12 =6 units
12 =0 .5 units
The next step is to determine the height of each rectangular strip. In this case, the right Riemann
sum is applied meaning that the rectangles are touching the curve y=2 x √x2 +1 with the right
hand corners (Khan Academy, 2018). The heights of the 1st, 2nd, 3rd ...12th rectangular strips are
the function values at the corresponding x value. The heights of the four rectangular strips are
calculated as follows:
x y=2 x √ x2 +1
0.5 1.1
1 2.8
1.5 5.4
2 8.9
2.5 13.5
3 19.0
3.5 25.5
4 33.0
4.5 41.5
5 51.0
5.5 61.5
6 73.0
The area of each rectangular strip is calculated by multiplying the width of the strip with the
height of that particular trip as follows:
Strip # Area (square units)
1 0.5 x 1.1 = 0.6
2 0.5 x 2.8 = 1.4
3 0.5 x 5.4 = 2.7

Area Under the Curve 3
4 0.5 x 8.9 = 4.5
5 0.5 x 13.5 = 6.7
6 0.5 x 19.0 = 9.5
7 0.5 x 25.5 = 12.7
8 0.5 x 33.0 = 16.5
9 0.5 x 41.5 = 20.7
10 0.5 x 51.0 = 25.5
11 0.5 x 61.5 = 30.7
12 0.5 x 73.0 = 36.5
The approximate area under the curve is obtained by adding the areas of the 12 rectangular strips
as follows (TechnologyUK, 2018):
Approximate area = 0.6 + 1.4 + 2.7 + 4.5 + 6.7 + 9.5 + 12.7 + 16.5 + 20.7 + 25.5 + 30.7 + 36.5 =
168 square units.
Approximate area when n2 = 4
When n = 4, it implies that there are four rectangles between x = 0 and x = 6. The width of the
four rectangles is the same and calculated as follows ( 6−0 ) units
4 =6 units
4 =1.5 units
Next is to calculate the height of each rectangle. The right Riemann sum is also applied in this
case meaning that the rectangular strips are touching the curve y=2 x √x2 +1 with the right hand
corners. The heights of the first, second, third and fourth rectangular strips are the function
values at the corresponding x value. The heights of the four rectangular strips are calculated as
follows:
x y=2 x √ x2 +1
1.5 5.4083
3 19.0
4.5 41.5
6 73.0