Here are the essential concepts you must grasp in order to answer the question correctly.
Riemann Sums
Riemann sums are a method for approximating the value of a definite integral by dividing the area under a curve into smaller rectangles. The sum of the areas of these rectangles provides an estimate of the integral's value. Depending on whether the left or right endpoints of the subintervals are used, the Riemann sum can either overestimate or underestimate the actual area, which is crucial for understanding the behavior of the integral.
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Introduction to Riemann Sums
Definite Integrals
A definite integral represents the net area under a curve between two specified limits, in this case, from 0 to Ο/2 for the function cos(x). It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. Understanding definite integrals is essential for evaluating the total accumulation of quantities, such as area, over an interval.
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Overestimation and Underestimation
In the context of Riemann sums, overestimation occurs when the sum of the areas of the rectangles exceeds the actual area under the curve, while underestimation occurs when the sum falls short. For a decreasing function like cos(x) on the interval [0, Ο/2], the left Riemann sum will overestimate the integral, and the right Riemann sum will underestimate it. Recognizing these patterns is vital for accurately interpreting the results of numerical integration.
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Left, Right, & Midpoint Riemann Sums Example 1
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