Exploring accuracy of methods of approximating definite integrals I am planning on writing a paper on the accuracy of methods of approximation of definite integrals. Right now, I am planning to show the convergence of the midpoint rule, trapezoid rule and simpsons rule on a specific interval and function. Then, I will derive the formulas for their error bounds.
I feel that my paper lacks originality. What other concepts can I bring into my paper to explore these 3 methods in depth? It would be great if someone could suggest any topics I could explore related to these 3 methods of integration or just recommend any books that do the same.
 A: Does your specific function have to be well behaved?  What happens if the specific function is discontinuous (perhaps infinitely many times)?
Your three numerical integration techniques are approximation by rectangles, trapezoids, and whatever the region is that is bounded by the set difference of a rectangle and a parabola with axis parallel to an axis of the rectangle.  Why not use other bounding objects: circles, half-parabolas (so that the vertex of the parabola is on the left or right edge of the partition element), other shapes?
Could you find a way to estimate the local error, so you could improve the error by subdividing the partition where the error is large?  Can you use this to initially pick the widths of the partition elements?
Integrals of volumes over areas?  Integrals of heights over a path in the plane?  Integrals of temperatures along a path in space?  Similar generalizations to higher dimensional backgrounds?
What if the integrand is complex-valued?
Honestly, look at each piece of your recipe for numerically estimating an integral and ask "what radically different thing could I replace this with and still make sense of what is happening?"
