Table of Contents
Explain in Detail :Important Riemann Sum Terms
A Riemann sum is a way to approximate the area under a curve by adding up the areas of many small squares. The approximation gets better as the squares get smaller.
There are three important terms used in Riemann sums:
– The function being approximated is called the “integrand.”
– The area of a square is called the “height” or “dx.”
– The number of squares used to approximate the area is called the “numerator.”
Riemann Sum
A Riemann sum is a method for approximating the area under a curve. The curve is divided into a number of sections, and the area of each section is calculated. The total area is then approximated by adding up the individual areas.
Riemann Integral Formula
The Riemann integral formula states that the integral of a function can be found by breaking the function into a sequence of pieces, each of which is approximated by a Riemann sum.
Properties of Riemann Integral
1. The Riemann integral is a function from a subset of the real numbers to the real numbers.
2. The Riemann integral is defined as the limit of a sequence of Riemann sums.
3. The Riemann integral is a linear operator.
4. The Riemann integral is a positive operator.
5. The Riemann integral is a monotone operator.
Riemann Sum Example
The following table gives the area of a rectangle under a curve for six different points along the curve.
\begin{array}{ccc}
x & y & A \\
1 & 1.5 & 3.75 \\
2 & 2.5 & 5.625 \\
3 & 3.5 & 7.5 \\
4 & 4.5 & 9.375 \\
5 & 5.5 & 11.25 \\
6 & 6.5 & 13.125
\end{array}
We can approximate the area under the curve by using a Riemann sum. In this example, we will use the leftmost point, the rightmost point, and four points in between.
\begin{array}{ccc}
x & y & A \\
1 & 1.5 & 3.75 \\
2 & 2.5 & 5.625 \\
3 & 3.5 & 7.5 \\
4 & 4.5 & 9.375 \\
5 & 5.5 & 11.25 \\
6 & 6.5 & 13.125
\end{array}
The Riemann sum for this example is:
\begin{align}
A &= 3.75+5.625+7.5+9.375+11.25+13
