In mathematics, the midpoint rule approximates the vicinity between the graph of the feature f(x) and the x-axis by using including the regions of rectangles with midpoints which are factors on f(x).
To find the place for specific rectangles and for a preferred n we get:
$$ \int_a^b f(x) \, dx = \Delta{x} \left( f\left(\frac{x_0 + x_1}{2}\right) + f\left(\frac{x_1 + x_2}{2}\right) + f\left(\frac{x_2 + x_3}{2}\right) + \ldots + f\left(\frac{x_{n-2} + x_{n-1}}{2}\right) + f\left(\frac{x_{n-1} + x_n}{2}\right) \right) $$
however, an online midpoint rule calculator can remedy features to approximate the integrals the usage of this formulation right away whilst you enter the upper and lower limits.
Midpoint rule Example:
Find the midpoint rule for \( \int_1^5 \sqrt{x^2 + 3} \, dx \), where the number of rectangles is 4.
Solution:
The integral \( \int_1^5 \sqrt{x^2 + 3} \, dx \) with n = 4 using the midpoint rule.
The midpoint rule formula is:
$$ \int_a^b f(x) \, dx = \Delta{x} \left( f\left(\frac{x_0 + x_1}{2}\right) + f\left(\frac{x_1 + x_2}{2}\right) + f\left(\frac{x_2 + x_3}{2}\right) + \ldots + f\left(\frac{x_{n-2} + x_{n-1}}{2}\right) + f\left(\frac{x_{n-1} + x_n}{2}\right) \right) $$
Where \( \Delta{x} = \frac{b - a}{n} \)
We have \( a = 1 \), \( b = 5 \), \( n = 4 \).
So, \( \Delta{x} = \frac{5 - 1}{4} = 1 \).
Divide the intervals [1, 5] into n = 4 subintervals with the length \( \Delta{x} = 1 \) for the following endpoints:
A = 1, 2, 3, 4, 5 = b
A midpoint rule approximation calculator can approximate the correct place under a curve among one-of-a-kind factors.
Now, decide the function at the factors of the subintervals.
$$ f\left(\frac{x_0 + x_1}{2}\right) = f\left(\frac{1 + 2}{2}\right) = f(1.5) = \sqrt{(1.5)^2 + 3} = 2.9580 $$
$$ f\left(\frac{x_1 + x_2}{2}\right) = f\left(\frac{2 + 3}{2}\right) = f(2.5) = \sqrt{(2.5)^2 + 3} = 3.5355 $$
$$ f\left(\frac{x_2 + x_3}{2}\right) = f\left(\frac{3 + 4}{2}\right) = f(3.5) = \sqrt{(3.5)^2 + 3} = 4.1389 $$
$$ f\left(\frac{x_3 + x_4}{2}\right) = f\left(\frac{4 + 5}{2}\right) = f(4.5) = \sqrt{(4.5)^2 + 3} = 4.6077 $$
Now, add the values and multiply by \( \Delta{x} = 1 \). So,
$$ 1 \times (2.9580 + 3.5355 + 4.1389 + 4.6077) = 15.2401 $$
The midpoint approach is more particular than the trapezoidal technique. this is recommended via the composite error bounds, but they do now not rule out the possibility that the trapezoidal may be more correct in some cases.
when the underlying feature is clean, the trapezoidal rule isn't as correct as the Simpson rule because the Simpson rule makes use of a quadratic approximation as opposed to a linear approximation. The method is generally given for an abnormal range of factors.
The Riemann Sum Midpoint requires our estimation of the function. We integrate it on the midpoint of each C program language period and use those values to find the heights of various rectangles affecting this purpose.
