Question

question 5
which of these describes the definite integral of a continuous function ( f ) on the interval (a,b)?
the limit of a riemann sum as the number of subdivisions (rectangles) goes to zero.
the limit of a riemann sum as the number of subdivisions (rectangles) goes to infinity.
the correct answer is not listed.
the average of the left and right riemann sums
a left or a right riemann sum

Explanation:

The definite integral of a continuous function \( f \) on \([a,b]\) is defined as the limit of a Riemann sum as the number of subdivisions (rectangles) \( n\) goes to infinity. A left - or right - Riemann sum is just an approximation. The average of left and right Riemann sums is also an approximation. And when the number of subdivisions (rectangles) goes to zero, the width of each rectangle \(\Delta x=\frac{b - a}{n}\) would go to infinity (since \(n\to0\)), which is not the correct concept for the definite integral.

Answer:

The limit of a Riemann sum as the number of subdivisions (rectangles) goes to infinity.