I am having trouble deciphering the opening gambit of an explanation of mean values of functions. It begins as follows: "Consider the part of the curve y = f(x) for values of x in the range a ≤ x ≤ b." A graph is shown with a curve cutting the x axis at c with a shaded positive area bounded by the curve and the line x=a to the left of c and a shaded negative area bounded by the curve and the line x = b to the right of c. "The mean value of y in this range is the average value of y for that part of the curve. The sum of the ordinates (ie values of y) between x= a and x = c occupies the shaded area above the x axis and is positive. This area is ∫acy dx Hence the sum of the ordinates between x = a and x= c is ∫acy dx" I understand that an ordinate is the value of y. But are the ordinates taken at integer values of x or continuous values of x. I don't see how the sum of ordinates is equal to the value of the area under the curve. I must have misunderstood the definition of the sum of the ordinates. I can see how the sum of the continuous ordinates multiplied by change in x as change in x goes to 0 might equal the area under the curve. Sorry if my terminology and description of the graph leave a lot to be desired.