MATH 19B Lecture 5: Math 19B 10_03_2016

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 1に.) 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

000 00 all WorR neatly on separate paper, which should be stapled to this page. e the limit process and summation to find the area betweenf(x) +15, x-1, and x 3, and the x-axis. a) Assuming there are n rectangles in the interval betweenx-1 and x-3, what is the width of each one? b) Write the formula representing the left endpoint of each rectangle. c) Write the formula representing the right endpoint of each rectangle. d) Use either the left or right endpoint (your choice) to represent the height of each rectangle, using your given function. e) Write the summation notation representing the area of n rectangles to approximate the area under f, using your answers to (a) and (c). f) Use the summation formulas as necessary to evaluate the summation. Your answer will be a formula that will give the area of for any number of rectangles, n' (8 points) s nd the area by evaluating the limit as the number of rectangles goes to infinity of your answer to (f). (2 points) ite the formula for the same area as a definite integral, then use the Fundamental Theorem of Calculus to ate the integral. (5 points)

x-y+1 1-2 -1 -3 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. -it-y 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 divisions, cover the region with the appropriate number of rectangles, and find the approximate area. 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 (e), 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. Hnt: Think of rotating the region R. e) Evaluate the integral you derived in part (d) using the fundamental theorem of calculus. (f) Calculate the exact value of 1 using Riemann sums. (g) Are your answers for the previous two parts equal? ζ