I am asked to deduce the shape of a curve by knowing the following: The estimate of the definite integral for the area using the trapezium rule with 2 intervals of equal widths is above the real value. The estimate of the definite integral for the area using the trapezium rule with 4 intervals of equal widths is below the real value. Can anyone help with this problem? Thanks
I believe you mean the Trapezoid Rule. Consider that this method estimates the definite integral for a function by placing points on the curve for f(x) and essentially 'connecting the dots' with straight lines. You will have a 'dot' at the start of the interval for integration, some dots in between, and one at the end of the interval. So you're looking for a shape for the f(x) curve so that when one dot is added on the curve at the midpoint of the interval, the area of the trapezoids created exceeds the area under the curve. Yet, when dots are also added at the quarter-intervals, the total area of all four trapezoids is now less than the area under the curve. Try out some curve shapes between the two endpoints and see what might fit this description...