I was given a decreasing concave down graph of the integral F(x) from 0-8.

i have to say whether the following are greater then, less then,equal to, or unable to determine each other

1. T (greater then) M
2. T (greater then) R
3. M (less then) S
4. S ? actual integral

where
t=trapazoidal rule
m=midpoint
s=simpson
r=right end points

are these right?
I said you weren't sure with simpsons whether it was greater than or less than the integral so it would be a ?

the others were determined based on increasing/decreasing and concavity

To determine whether one approximation method is greater than, less than, equal to, or unable to determine in relation to another approximation method, we can use the concept of the area under a curve.

1. T (greater than) M: The trapezoidal rule approximates the area under the curve by dividing the interval into trapezoids and summing their areas. The midpoint rule approximates the area under the curve by dividing the interval into rectangles and summing their areas. In general, the trapezoidal rule tends to give a better approximation than the midpoint rule, so T is greater than M.

2. T (greater than) R: The trapezoidal rule approximates the area under the curve by including both the left and right endpoints of each interval. The right endpoint rule approximates the area by considering only the right endpoints. When the graph is decreasing and concave down, the right endpoint rule tends to give an overestimate of the area, while the trapezoidal rule is closer to the actual value. Therefore, T is greater than R.

3. M (less than) S: The midpoint rule approximates the area under the curve by considering the midpoint of each interval. The Simpson's rule approximates the area by using a second-degree polynomial to fit the curve within each interval. The Simpson's rule tends to give a better approximation than the midpoint rule, especially for curves that are not linear. Therefore, M is less than S.

4. S ? actual integral: It can be difficult to determine whether the Simpson's rule is greater than or less than the actual integral for a decreasing concave down curve. Without more specific information about the graph and the function being integrated, it is not possible to accurately determine the relationship between S and the actual integral. Hence, the relationship between S and the actual integral is undetermined.

Remember that these comparisons are general observations based on the properties of the approximation methods under specific conditions. The actual values may vary depending on the specific shape of the graph and the function being integrated.