The A option is correct.
(a) It is given that the function f is a constant function on the interval [a,b]
and the right and left Reimann sum give the exact value of
for any n. Choose the correct statement as follows.
The area under a constant function is a rectangle, so the rectangles of a Riemann sum cover exactly the whole area.
This means that the right and left Riemann sums give the exact value of for any n, regardless of how large or small n is. So, option A is correct.
B. The left and right Riemann sums only give an exact value for values of n that are very large. This statement is false because the left and right Riemann sums give the exact value of f(dx) for any n , regardless of how large or small n is. This is because the area under a constant function is a rectangle, so the rectangles of a Riemann sum cover exactly the whole area.
So, the option B is false.
C. The Riemann sum gives an approximation of an integral and never an exact value. This statement is false because, in the case of a constant function, the Riemann sum does give an exact value of the integral. This is because the area under a constant function is a rectangle, so the rectangles of a Riemann sum cover exactly the whole area.
So, the option C is false.
D. Only a midpoint Riemann sum will give an exact value of the integral. This statement is false because both left and right Riemann sums give an exact value of f(dx) offer regardless of how large or small n is. This is because the area under a constant function is a rectangle, so the rectangles of a Riemann sum cover exactly the whole area.
So, the option D is false.
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