Question

Let f(x) = 14 − 2x. (a) Sketch the region R under the graph of f on the interval [0, 7]. Use a Riemann sum with five subintervals of equal length (n = 5) to approximate the area (in square units) of R. Choose the representative points to be the right endpoints of the subintervals. square units (c) Repeat part (b) with ten subintervals of equal length (n = 10). square units (d) Compare the approximations obtained in parts (b) and (c) with the exact area found in part (a). Do the approximations improve with larger n? Yes No

Answer 1

Q+ f(x) = 14-2x. (0,7) Area under Carne- (0,14) 7 A $ f(x) dx (7,0) 7 (14-22) de 7 = (142-x², 14x7 - 49 49 Aus Reimann sum Ax b-a f(x) dx. b n=5. f(x) du J. (14-2x) dx. 7-0 Ax= 70 Na s Reimann right sum AR = AX I Hlavax) + + f(a+20x) + f(a+3 Ax) -t f(b-ox)

5 As - 공 () + (2²) + }(3Q) + (4) ++(s) = + + + + 2 7 1 1 x22 TH 39.2 Aug na 10 A= 7 ) 70 AR Aa Ala + ax) + f (a + 2.ax) + 4 (A+3Ax) 4 45-axy) - 금 (금) + 4 (2) + ? (5) • (금) . … (4) * (1)) J [63] 94.1 Aug
As correct area(A) is 49

With n=5 we got riemann sum 39.2

And with n=10 we got riemann sum 44.1

So we conclude as we increase the number of subintervals we get more accurate answers.