Connecting the Limit of a Sequence of Riemann Sums to Antiderivatives
We can compute the exact area of the region below the graph of a function by calculating the Riemann sum based on \(n\) subintervals, and then computing the limit of that expression as \(n \to \infty\). However, this process can be tedious to complete by hand. In the exercise below, you will explore an alternate way to determine the area of the region.
3. For the function \( f(x) \) assigned by your instructor in the lab report, answer the following questions.
(a) Find the antiderivative \( F(x) \) of \( f(x) \). In other words, find a function \( F(x) \) such that \( F'(x) = f(x) \) for all \(x\) in the given interval. To check your answer, use the SageMath evaluation cell below to check that your choice for \( F(x) \) has the desired property. (Recall that SageMath‘s command diff(F,x) computes the derivative of \( F(x) \).)
(b) Use the SageMath evaluation cell below to evaluate \( F(b) – F(a) \) for the values of \( a \) and \( b \) given by your instructor. Record this in your lab report. How does the answer you get here compare to the estimate of the limit of Riemann sums that you made in Exercise 2?