MATH115 Lecture Notes - Lecture 1: Emule, Product Rule, European Cooperation In Science And Technology

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

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

2/9x2 + 25 dx Determine the function f(x) given that f'(x) = 1 + sin x and f(0) = 3 The acceleration due to gravity on the moon is 1.62 m/s2 . If a ball is thrown from the surface of the moon upwards with a velocity of 20 m/s, what will its maximum height be? Estimate the net area under the graph of y=x sin x between x = 0 and x = 25 using a Right Riemann Sum with 100 subintervals. Find the average value of f (x)=sin (x) between 0 and 2 pi using the formula involving the integral. Explain why your answer makes sense by considering the graph of y=sin x Show all your work. Evaluate the following integrals Albumin is a protein found in the blood. A common test to see if a person has liver disease is to check that person's albumin levels. If the concentration of albumin in the blood serum is under 32 mg/ml, then liver disease is suspected, as liver disease is the most common cause of low albumin. This test is performed by taking a 1 mL sample of blood serum and running it through an HPLC machine which separates the albumin from the other chemicals in the blood. The output of one such test is displayed below. Does this person have low albumin? HINT: Use the trapezoid method to determine the area under this curve = weight of albumin in the 1 mL sample of blood serum Given that f(x)dx =8 and f(x)dx=-2 c, evaluate the following (f{x)+2x +1)dx 3 f[x)dx The density 'p' of a certain cylindrical rod that has radius 0.1 metres and length 4 metres varies along the length x of the rod by the formula p(x)=3+ 2 x kilograms per cubic meter for 0 x 4. Find the total mass of the rod. HINT: How much weight is there in the differential slice at "x"? It is p(x) times the volume of the slice (pi r2 dx )