Interpretation of definite integrals

Question

Integration can be understood in many ways. Describe two ways we can interpret the result of a definite integral. Given a, and b (The limits) are real numbers. Explain why these interpretations make sense.

Answer 1

The basic interpretation of a definite integral is covered in the early part of a Calculus course. For a continuous function f(x) that is never negative on an interval [a,b], the definite integral from a to b  represents the area under the curve, above the x-axis, over that interval.

 

This interpretation make sense owing to the definition of the limit of Riemann sums, where (essentially) the areas of infinitely many very skinny rectangles of width dx and height f(x) are accumulated.