Midpoint Approximation
Midpoint approximation is a method used to estimate the value of a definite integral using rectangles
Midpoint approximation is a method used to estimate the value of a definite integral using rectangles. It is also known as the midpoint rule or rectangular rule. This method is an application of Riemann sums and is especially useful when the function being integrated is not easily integrable, or when exact values are not available.
To use the midpoint approximation, follow these steps:
1. Divide the interval [a, b] into n equal subintervals. The length of each subinterval, denoted by Δx, is given by Δx = (b – a) / n.
2. Find the midpoint of each subinterval. The midpoint is calculated as follows: mid_i = a + (i – 1/2)Δx, where i ranges from 1 to n.
3. Evaluate the function at each midpoint. Calculate f(mid_i) for each subinterval.
4. Multiply each function value by the length of the subinterval, Δx. Find the area of the rectangle formed by multiplying f(mid_i) and Δx.
5. Sum up the areas obtained in step 4 for all the subintervals. This sum represents an estimation of the definite integral of the function over the interval [a, b].
Mathematically, the midpoint approximation formula is given by:
Approximation = ∑[i=1 to n] f(mid_i) * Δx
This formula represents the sum of all the areas of the rectangles formed by multiplying the function values at the midpoints by the length of the corresponding subintervals.
It is important to note that as the number of subintervals, n, increases, the approximation tends to become more accurate. The accuracy of the approximation can also be improved by selecting the subintervals in a way that covers the curve more closely.
However, it is crucial to remember that the midpoint approximation is an approximation method and not an exact value. It serves as a good estimate when other methods are difficult to apply, but it may not yield the exact result.
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