Thread: Clarification over the Definition of the Definite Integral

I'm currently taking my third semester of calculus, and it's somewhat saddening that I don't quite understand this definition.

For single variable calculus, I get that you divide the segment between the limits of integration into pieces, and then pick a random point within each for the function value. The products of the length of the segments and the function values create Riemann rectangles that approximate the area under the curve.

What I don't understand is why the definite integral's value is the limit as the biggest segment (the norm) approaches zero. Doesn't that segment just disappear, leaving you with what you originally have, except with that specific rectangle removed? How does this better approximate the integral?

Thanks for any input.

Originally Posted by MathTooHard

I'm currently taking my third semester of calculus, and it's somewhat saddening that I don't quite understand this definition.

For single variable calculus, I get that you divide the segment between the limits of integration into pieces, and then pick a random point within each for the function value. The products of the length of the segments and the function values create Riemann rectangles that approximate the area under the curve.

What I don't understand is why the definite integral's value is the limit as the biggest segment (the norm) approaches zero. Doesn't that segment just disappear, leaving you with what you originally have, except with that specific rectangle removed? How does this better approximate the integral?

Thanks for any input.

It's not the limit of the Riemann sum as you take the largest single rectangle and shrink its width down to zero. It's the limit of Riemann sums as you make ALL of the rectangles stay below a smaller and smaller width, i.e., the limit of a sequence of Riemann sums where the width of the biggest rectangle of approaches zero as n goes to infinity.

My textbook says that the largest rectangle's width should go to zero. How does this cause the other rectangles' widths' to approach zero? Also, as the width(s) decrease, how are new rectangles created (such that there are more and more rectangles)?

Originally Posted by MathTooHard

My textbook says that the largest rectangle's width should go to zero. How does this cause the other rectangles' widths' to approach zero? Also, as the width(s) decrease, how are new rectangles created (such that there are more and more rectangles)?

You're not decreasing the width of a rectangle in a single partition. You're picking a sequence of entirely new partitions of rectangles, such that the mesh (the length of the largest interval in the partition) gets successively closer and closer to zero as you go further into the sequence. When they say "as the largest rectangle's width approaches zero," they mean "as you choose finer and finer-grained partitions." The limit of the Riemann sums of the partitions is the integral.