Derive the formula of integration using Riemann sum? Estimate the area under f(x) = x3over the interval [0,8] usig the mid-point Riemann Sum for n = 4.
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Given a function f(x) where f(x)≥0 over an interval a≤x≤b, we investigate the area of the region that is under the graph of f(x) and above the interval [a,b] on the x-axis. For example, the below purple shaded region is the region above the interval [−1,10] and under the graph of a function f. Such an area is often referred to as the “area under a curve.”
Since the region under the curve has such a strange shape, calculating its area is too difficult. But calculating the area of rectangles is simple. Let's simplify our life by pretending the region is composed of a bunch of rectangles. To turn the region into rectangles, we'll use a similar strategy as we did to use Forward Euler to solve pure-time differential equations.
As illustrated in the following figure, we divide the interval [a,b] into n subintervals of length Δx (where Δx must be (b−a)/n). We label the endpoints of the subintervals by x0, x1, etc., so that the leftmost point is a=x0 and the rightmost point is b=xn. The picture shows the case with four subintervals.
Answer:
Derive the formula of integration using Riemann sum? Estimate the area under f(x) = x3over the interval [0,8] usig the mid-point Riemann Sum for n = 4.