Here are the essential concepts you must grasp in order to answer the question correctly.
Midpoint Rule
The Midpoint Rule is a numerical method used to approximate the value of a definite integral. It involves dividing the interval of integration into 'n' subintervals of equal width and using the midpoint of each subinterval to calculate the area of rectangles that approximate the area under the curve. The formula for the Midpoint Rule is given by: M_n = Δx * Σ f(x_i*), where Δx is the width of each subinterval and x_i* is the midpoint of the i-th subinterval.
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Definite Integral
A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is denoted as ∫(a to b) f(x) dx and can be interpreted as the limit of Riemann sums as the number of subintervals approaches infinity. The definite integral provides a way to calculate total accumulation, such as distance, area, or volume, depending on the context of the function.
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Subintervals
Subintervals are smaller segments into which the main interval of integration is divided when applying numerical methods like the Midpoint Rule. The number of subintervals, denoted as 'n', determines the accuracy of the approximation; more subintervals generally lead to a better approximation of the integral. The width of each subinterval is calculated as Δx = (b - a) / n, where [a, b] is the interval of integration.
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