Question

The table gives the values of a function obtained from an experiment. Use them to estimate 9 f(x) dx 3 using three equal subintervals with right endpoints, left endpoints, and midpoints. x 3 4 5 6 7 8 9 f(x) −3.4 −2.2 −0.6 0.2 0.8 1.5 1.7

Answer 1

With right endpoints: -6.4

With left endpoints: 3.8

With midpoints: -1

Explanation:

Given

x 3 4 5 6 7 8 9

f(x) −3.4 −2.2 −0.6 0.2 0.8 1.5 1.7

We are asked to estimate the integral 3 to 9 f(x) dx.

Using the intervals 3 to 5, 5 to 7 and 7 to 9, the length of the intervals is Δx = 2

With right endpoints the integral is:

f(3)*Δx + f(5)*Δx + f(7)*Δx = (-3.4)*2 + (-0.6)*2 + 0.8*2 = -6.4

With left endpoints the integral is:

f(5)*Δx + f(7)*Δx + f(9)*Δx = (-0.6)*2 + 0.8*2 + 1.7*2 = 3.8

With midpoint s the integral is:

f(4)*Δx + f(6)*Δx + f(8)*Δx = (-2.2)*2 + 0.2*2 + 1.5*2 = -1

Answer 2

We shall use 3 equal subintervals, each with width = 2.

Integral with left end points:
A-left = -3.4*2 + (-0.6)*2 + 0.8*2 = -6.4

Integral with right end points:
A-right = -0.6*2 + 0.8*2 + 1.7*2 = 3.8

Integral with midpoints:
A-mid = -2.2*2 + 0.2*2 + 1.5*2 = -1.0

Notice that A-left < A-mid < A-right.