Mastering the Midpoint Rule: A Comprehensive Guide to Approximating Definite Integrals in Calculus

Midpoint Rule

The midpoint rule is a method used in calculus to approximate the value of a definite integral

The midpoint rule is a method used in calculus to approximate the value of a definite integral. It is based on dividing the interval of integration into subintervals and using the midpoint of each subinterval as the representative value for that interval.

To apply the midpoint rule, follow these steps:

1. Determine the interval of integration: The definite integral is usually given in the form ∫[a, b] f(x) dx, where ‘a’ and ‘b’ are the lower and upper limits of integration, respectively.

2. Divide the interval into subintervals: Choose the number of equally spaced subintervals ‘n’ you want to use to approximate the integral. The larger the value of ‘n’, the more accurate the approximation will be. Each subinterval width (Δx) will be (b – a)/n.

3. Find the midpoint of each subinterval: For each subinterval, determine the midpoint by adding half of the subinterval width (Δx/2) to the lower limit a. The midpoint of the i-th subinterval will be denoted as xi.

4. Evaluate the function at each midpoint: Calculate the value of the function f(xi) at each midpoint.

5. Compute the average of the function values: Add up all the function values obtained in the previous step and divide the sum by the number of subintervals (n). This will give you the average function value.

6. Multiply the average function value by the subinterval width: Multiply the n-averaged function value by the subinterval width (Δx) to get an approximation of the integral of the function over the interval [a, b].

Mathematically, the midpoint rule approximation for the integral can be written as:
∫[a,b] f(x) dx ≈ Δx * (f(x1) + f(x2) + … + f(xn)),

where Δx = (b – a)/n, and x1, x2,…, xn are the midpoints of each subinterval.

It’s important to note that the midpoint rule is an approximation method and may not provide the exact value of the integral. However, as the number of subintervals increases (as ‘n’ approaches infinity), the approximation approaches the exact value of the integral.

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