The value of the definite integral [tex]\int\limits^{10}_{0} {f(x)} \, dx[/tex] has an approximate value of 5 units.
A definite integral within a given interval is represented graphically by the net area below the curve. In this question we must estimate the total area of the curve by right Riemann sum. The most accurate approximation is using Riemann sum with trapezoids, whose formula is defined below:
[tex]A = \frac{b-a}{2\cdot n} \sum_{i=0}^{n-1} \left[f(x_{i})+f(x_{i+1})\right][/tex] (1)
Where:
If we know that n = 5, a = 0 and b = 10, then the area of the curve is approximately:
[tex]A = \left[\frac{10-0}{2\cdot (5)} \right]\cdot [(f(0)+f(2))+(f(2)+f(4))+(f(4)+f(6))+(f(6)+f(8))+(f(8)+f(10))][/tex]
[tex]A = f(0) + 2\cdot f(2) + 2\cdot f(4) + 2\cdot f(6) + 2\cdot f(8) + f(10)[/tex]
[tex]A \approx 3 + 2\cdot (0) + 2\cdot (-1) + 2\cdot (-2)+2\cdot (2) + 4[/tex]
[tex]A\approx 5[/tex]
The value of the definite integral [tex]\int\limits^{10}_{0} {f(x)} \, dx[/tex] has an approximate value of 5 units. [tex]\blacksquare[/tex]
The figure of the function f(x) is missing. We include a simplified version of the image in the picture attached below. In addition, the statement is poorly formatted, correct form is shown below:
Estimate [tex]\int\limits^{10}_{0} {f(x)} \, dx[/tex] using five subintervals with the following.
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