Question

Peer Leading Exercise 7 Spring 2019: Area Under the Given a function (x), the area under the curve is the area of the region bordered by the x -sxis and the graph of y(x). Area under the curve is somehow related to anti-derivatives. We wish to Example: Let f(x) -10-2x. Find the area under the curve between x 0 and x graph to help you visualize what is going on. Do you recognize the shape? 5. We include a 2 2 What is the area of the region bounded by the graph of f) 10- 2x and the x-axis betweenx and x = 5? You should explain how you are getting your value! Find an antiderivative for f(x). This means: Find a new function F(x) so the derivative FC) is equal to f(x). In this case specifically that means find a function F(x) with derivative F(x) 10-2x Evaluate F(x) atx5. Compare this value to the area you found above. Are they close? Find the area of the region bounded by the graph of f(x) and the x-axis between x0 and x 3 (it's a trapezoidI). Compare this area to the value F(3). Are they close as well Example: Let g)16-x Find the area under the curve between x and a graph to help you visualice what is going on Estimate the area under the curve by filing the region with 4 rectangles! The height of the rectangle wi be given by the value of the function (at one of the endpoints of the rectanglesl and the width of each rectangle will be 1. You should gooe "Reimann sumst and look at the images if you cannot picture what is going on What is your area estimate using the left end points? What is your area estimate using the right end points? What is the average of your two estimates? Find an anti-derivative, G(x) for the function g(x). Also, evaluate G(x) at x4. Compare this value with the average af your estimates. Are they clase?

Answer 1

Retween x-0 and x-5, we first divide this interval in to n equal parts. The width of each sub-interval would be 5/n. Now, the area between the curve f(x)-10-2x and the x axis can be n rectangles each of width 5/n. The area of all such rectangles would constitute the area between f(x) and the x axis As shown in the figure above the first rectangle, starting with x-0 would have a width of 5/n and length 10-2(5/n)- 10- 10/n Width of second rectangle would be same but its length at the right end point would be 10-2(10/n- 10- 20/n and so on. Area of all such rectangles when added would sum u as follows 5/n [(10-10/n) +(10- 20/n) 10-10n/n)] 5/nl(10(n)-10(1/2n n/n) 5/nl 10n-10/n( 1+2 .) 5/n [10n-10/m * n((n+ 1)2 5/n[10n-10(n+ 1)/21-60-25(n+1)/n=50-25-25/n The exact area worked out by the above method would be when M* This would be 50-25 25 units sum of 1st n natural number equals n(n+1%2

ox Aniderivative of t) -102, F)0-L) The area bounded by fx) and x axis, from x-0 tox-3 would be the area of the trapezoid as shown below 0