Question

Given the following function, f(x) xe * on [-1,1]: 7.1. Approximate the net area bounded by the graph of f and the x- axis on the interval using a left, right, and midpoint Riemann sum with n = 4. (8 Marks) 7.2 Sketch the graph of the function and show which intervals of [a, b] make positive and negative contributions to the net area. (2 Marks) [Sub Total 10 Marks]

Answer 1

h=u'[!'1-Juo k? R=(R) } 5-9 azt, 621 2 2 2 4 / 6 * (-)-1 an= 2 n 4 Sub Intuvala 0.5 -I, -05, 5,1 - X -1 -0.5 0.5 f(x) 1-2.7182 -018243 0.3678 3032 + Area z Lett subintuvals -1,-0.5, 0, 0.5 Area = sfadze sx [f(-) + +(-0.5)++10)+4(0.151) - Ste*dx = 4(-2.1182-018243+0+0-3033) l'aéudna -1.6196 Right Subintervals oors, a, ers; I Areas &cn) dza saf+4-0-5)+ f(0)+f(015)++(1) Ļ (-01$243 +0 +0-3032 +0.3678 ==(-0.82 824370.671) = Midpornit intervals -0.1533 0,07666 2 -10.5 moisto 2 2 itsoo sooto 2 2 2 0:25 0,75 -0.75 -0.25 0.1947 0.3542 f(x) : 0.5877 -0 13210 b Δί Areas I fealda [f(-0.95) + f(-0,25) + +0.25)+ f(0.75)) Ź (-1.5877 – 0.321070 -0.3210+ 0.1947+0.3542] a

=-0.6799 0 -1 1 x ēndx = {(-1-9087 +0.5687] = -1.3978 : juetes Svē”dx = -0.6799 negative Area s f(a) dx2 AX [f40.75)+41012) - $-4.5897 ---300] = (-1.9087] - jtaba con a {f(0.25) -0.25) +f60.7) -1 Area -0.9543 Positive Area = f(x)da 2 $(0.1947+0.35427 1 / f(a) da 20:27445 0 Total Alca jferdza fifcerent iterad 0 = -0,95-43 +0,2744 Area =-0.6799