1. In caleulus II, we often need to compute areas of regions that cannot be described simply as For instance, a typical problem may ask you to find the area of the region neath the graph of y = f(z)". pictured below. x=y+1 The goal of this workshop is to explore a way of computing the area of such regions. We will be using "horizontal" Riemann sums. Please be cautious and note that the method we will be exploring here is not the Riemann sums method we learned in class. Let I be the following integral. IO 10 Let R be the region described by I. (a) Sketch the region R (b) Approximate the area of R by dividing up the region horizontally and covering the region with two rectangles. That is, you should divide the interval of y-values into two equal subintervals. Then for each subinterval, draw a rectangle whose width is determined by the function values. (c) Now divide up the region horizontally into 10, 100, and 10,000 rectangles. For each number of _1 1ã«.) divisions cover the region with the appropnate number ofre tangles and find the appro nate area. o Express the approximate area using summation notation. Use a calculator to calculate these sums to three decimal places. (d) Based on your estimates from part (c), what do you think the area of R is? Write the sums you found in part (c) as the Riemann sums of some other function over some other interval. Thus find an alternative integral expression for the area of R. 1- Hint: Think of rotating the region R. S (e) Evaluate the integral you derived in part (d) using the fundamental theorem of calculus. (f) Calculate the exact value of I using Riemann sums. (g) Are your answers for the previous two parts equal? lo
