Approximation of the Area

Question

Find an approximation of the area of the region R under the graph of the function f on the interval [0, 2]. Use   n = 5 subintervals. Choose the representative points to be the midpoints of the subintervals

f(x) = x² + 2

Answer 1

The midpoints are at x = 0.2, x = 0.6, x = 1, x = 1.4, and x = 1.8

f(0.2) = 2.04

f(0.6) = 2.36

f(1) = 3

f(1.4) = 3.96

f(1.8) = 5.24

Area ≈ 2.04(0.4) + 2.36(0.4) + 3(0.4) + 3.96(0.4) + 5.24(0.4) = 0.4(2.04 + 2.36 + 3 + 3.96 + 5.24) = 6.64

Answer 2

the interval of 2 is divided by 5 to get 5 subintervals of 2/5 each. the midpoints are 1/5, 3/5, 1, 7/5 and 9/5

f(1/5) = .04 + 2 = 2.04

f(3/5) = 9/25 + 2 = 2.36

f(1) = 1 + 2 = 3

f(7/5) =49/25 + 2 = 3.96

f(9/5) = 81/25 +2 = 5.36

multiply each of the f(x) values by 2/5 or .4, then sum the 5 areas

or just add up the 5 f(x) values and then multiply by 2/5

16.72 times 2/5 = 16.72/5 = 6.688

to check if this is accurate, take the integral of x^2 + 2 evaluated at x=2

x^3/3+2x = 2^3/3+4 = 8/3+4 = 2 2/3 + 4 = 6 2/3 = 6.666....

the approximation is within less than 0.02 of the exact answer