If f is a linear function on the interval [a,b], then a midpoint Riemann sum gives the exact value of Integral from a to b f (x )dx for any n. Is the statement true or false? Explain.

Remember that the integral of a function is the area below the curve, even if your function is linear the mid point will only give you an approximation of the height. Here's a picture that depicts what I am talking about.

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rahulsharma789888 rahulsharma789888 · Answer 2 · 2021-12-06T09:27:45+01:00

If f is a linear function on the interval [a,b] then a midpoint Riemann sum gives the exact value of Integral

[tex]\rm \int\limits^b_a {f(x}) \, dx[/tex]

for any n. This statement is false.

Riemann sums only estimate the area under the curve to an approximate value.

For an decreasing function considering the left side of the points of domain the sum of areas of rectangles gives a value that is more than the integral of the function.

Similarly for an increasing function the Riemann sum' s approximation on left side of the points of domain gives underestimation.

For a linear function the Riemann sum approximation is attached in the figure.

Considering the left side of domain points on x axis the underestimation is shown by red color.

[tex]\rm \int\limits^b_a {f(x}) \, dx..........(1)[/tex]

Equation (1) shows area under the curve of function f(x) from domain point a to point b.

Hence the exact values are not calculated by Riemann's Sum approximation. So the given statement is false.

For more information please refer to the link given below

https://brainly.com/question/104442

If f is a linear function on the interval​ [a,b], then a midpoint Riemann sum gives the exact value of Integral from a to b f (x )dx for any n. Is the statement true or​ false? Explain.

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If f is a linear function on the interval [a,b], then a midpoint Riemann sum gives the exact value of Integral from a to b f (x )dx for any n. Is the statement true or false? Explain.