2 3 FIGURE 22
Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
Calculate the Riemann sum for f (x, y) = x - y and the shaded domain 1) in Figure 22 with two choices of sample points, • and o. Which do you think is a better approximation to the
A function:
To Find:
Reimann Sum for the given function and domain.with two choices of sample points.
Which one is a better approximation to the integral of f and D?
Consider:
function: .
The domain D is contained in the rectangle .
The rectangle R is partitioned into 12 subrectangles each with an area of .
The sample point indicated by are upper -right corners of the subrectangles are as follows:
.
The corresponding Reimann sum is calculated as follows:
The sample point indicated by are midpoints of the subrectangles is as follows:
.
The corresponding Reimann sum is calculated as follows:
The second approximation is the better approximation to the integral as more points are considered to compute it.
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