# Thread: another question on approximating integrations

1. ## another question on approximating integrations

Let M_n be the nth Midpoint Rule approximation for a function f.
Let T_n be the nth Trapezoidal Rule approximation.
Show that if f''(x) >= 0 (i.e. f is convex on [a,b]), then for any natural numbers m,n, we have
M_n <= integral f(x)dx <= T_m.
If f''(x) <= 0 on [a,b], this inequality is reversed.

I've been playing around with the respective error bounds forever. How do you make this work algebraically? I'm having trouble simplifying the expression and working out the absolute values.

2. Originally Posted by BrainMan
Let M_n be the nth Midpoint Rule approximation for a function f.
Let T_n be the nth Trapezoidal Rule approximation.
Show that if f''(x) >= 0 (i.e. f is convex on [a,b]), then for any natural numbers m,n, we have
M_n <= integral f(x)dx <= T_m.
If f''(x) <= 0 on [a,b], this inequality is reversed.

I've been playing around with the respective error bounds forever. How do you make this work algebraically? I'm having trouble simplifying the expression and working out the absolute values.
Draw a diagram showing just one of the intervals and the area corresponding
to each method's contribution to it's estimate of the integral.
Compare these with the actual area under the curve on the interval.

Now turn that into an analyitc proof.

RonL

3. Can you use the error inequalities? I'm trying that but I can't equate the different bounds and I get the integral times 2.