Before the Lab 1. Figure 1 shows the graph of a non-negative function fron the i

Question

Before the Lab 1. Figure 1 shows the graph of a non-negative function fron the interval la,bl. We want to write an expression for the sum of the areas of the four rectangles. We will call this sum R(4), and it will provide an estimate for the area under the graph of the functionfand above the interval [a,b] on the xaxis. The goal is to develop a formula that depends only on the function f and the interval endpoints a and b. a. Write an expression for the length of the interval [a,b] in terms of a and b. b. The four subintervals that form the bases of the rectangles along the x-axis all have the same length. This length is normally called Ax. Write a formula for Ar in terms of a and b. c. Note that xo a and that xi xo Ar Rewrite the expression for x as a formula involving only a and b (get rid of the xo and Ax in the formula). Also write down formulas for xa, x, and x, in terms of a and b. d. Note that the height of the first rectangle is given by f(x). Use the result of part (c) to rewrite this height in terms offand the endpoints a and b (eliminating x from the expression). a X

Explanation / Answer

R(4) = Sum of areas of 4 rectangles

The width of each rectangle = (b-a)/4 =delta x

Length of 4 triangles are = f(x1), f(x2) , f(x3) f(x4)

Hence sum of areas of 4 triangles = delta x [f(x1)+f(x2)+...+f(x4)]

= (b-a)/4 [f(a+(b-a)/4)+f(a+2(b-a)/4)+f(a+3(b-a)/4)+f(a+4(b-a)/4)]

= 0.25 (b-a)[f(a+(b-a)/4)+f(a+2(b-a)/4)+f(a+3(b-a)/4)+f(b)]

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For L(4) we take length as f(a), f(a+deltax)....f(a+3deltax)

Hence

L(4) = 0.25 (b-a)[f(a) +f(a+(b-a)/4)+f(a+2(b-a)/4)+f(a+3(b-a)/4)]

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