Riemann sums and the Fundamental Theorem of Calculus
Answer the questions below by using the graph above. You will need to enter the new function [math]f\left(x\right)=-\frac{x^2}{2}+2x+3[/math] in the box above and may have to grab the graph and lift it to move the dots on the x-axis to x=1 and x=5.
1. For the function [math]f\left(x\right)=-\frac{x^2}{2}+2x+3[/math] above on the interval [1,5], select n=4. [br](a) Is the left or right sum an upper bound for [math]\int_1^5f\left(x\right)dx[/math]?[br](b) For these Riemann sums, what is a bound on the integral?[br](c) Compute the Riemann sum for this case instead using the [b]midpoint[/b] of the interval as in our text.
2. Use the same function and interval as in question #1.[br](a) As you slide the values of n to be larger, what is true about the left and right Riemann sums?[br](b) When you slide n to its largest value, how accurately can you determine the definite integral? Be specific.
3. Use the fundamental theorem of calculus to explicitly solve [math]\int_{_1}^4\left(-\frac{x^2}{2}+2x+3\right)dx.[/math] Show all your steps.