A plot of \(f\) for inputs in the interval \([0,2]\) is shown in Figure 12.1.1. Break the interval \([0,2]\) into four equally sized subintervals and draw the rectangles that would be used to construct a Riemann sum to approximate the area under \(f\) on the interval \([0,2]\text{.}\) You can use whichever point you want on each subinterval to evaluate the height the of the rectangles.
Estimate the heights of the rectangles used in your Riemann sum above to estimate \(\displaystyle{\int_0^2 f(x)\enspace dx}\text{.}\) Write a couple of sentences about why you think your estimate for the definite integral of \(f\) is either an overestimate, an underestimate, or close to the true value.
Explain why it doesn’t matter what method (left endpoint, right endpoint, midpoint, etc.) you use for selecting which point is evaluated on each of the subintervals in the definition of the definite integral.
For each of the following functions plotted below, determine if the definite integral over the region shown will be positive, negative, or zero and write a sentence to justify your answer.
