Question

1. (lwt) Caleulate the desired approximation for the area between the curve and x-axis for the given interval. Include a sketch of the rectangles used in the approximation. USE EXACT VALUES OF TRIG FUNCTIONS Find Ls for f(x)-sin GJ on 1 0.5 0383 007 113 160723 20 2.667 2. (1wt) According to the USDA, the following table gives number of pounds of chicken the average person in the US consumed per year. Year 1995 2000 2005 2010 2015 Chicken consumed (bs/yt) 482 54.2 60.5 58.1 60.3 Suppose one of your parents is such an average person. Estimate the total amount of chicken they consumed from 1995 to 2015 using a right endpoint approximation. Include units with your final result.

Answer 1

The L_5 = Left Riemann sum(also Left Endpoint approx) therefore I = integral^b_a f(x)dx = delta x[f(x_0) + f(x_1) + ... + f(x_n)] where delta x = b - a/n \r\n Here, we have a = 0, b = 5/3, n = 5 Therefore delta x = 5/3 -0/5 = 1/3 Divide the interval [0, 5/3] into n = 5 subinterval of length delta x = 1/3 a = [0, 1/3], [1/3, 2/3], [2/3, 1], [1, 4/3], [4/3, 5/3] = b Now, L_5 = integral^b_a f(x)dx = delta x[f(x_0) + f(x_1) + f(x_2) + f(x_3) + f(x_4)] = 1/3[0 + 0.25 + 0.75 + 1 + 0.75] = 1/3[2.75] L_5 = 0.91667 Now for exact value I = integral^5/3_0 sin^2(pi/2 x)dx = integral^5/3_0 1 - cos^2(pi/2 x)/2 dx = 1/2 integral^5/3_0 [1 - cos pi x]dx = 1/2{[x]^5/3_0 - [sin pi x/pi]^5/3_0} = 1/2[5/3 - sin 5 pi/3/pi] = 0.97117 \r\n Here n = 4, Right endpoint approximation R_4 delta x = 2015 -1995/4 \r\n