Answer
(a) 4.2
(b) -6.2
(c) -0.8
Since the function is increasing, an estimate using the right endpoint of each subinterval is greater than the exact value of the integral.
Since the function is increasing, an estimate using the left endpoint of each subinterval is less than the exact value of the integral.
We can not know whether an estimate using the midpoint of each subinterval is less than, greater than, or exactly equal to the exact value of the integral.
Work Step by Step
(a) Using the information in the table, we can use three subintervals to estimate the value of the integral.
$\Delta x = \frac{b-a}{n} = \frac{9-3}{3} = 2$
We can use the right endpoint of each subinterval.
$x_1 = 5$
$x_2 = 7$
$x_3 =9$
$\sum_{i=1}^{3} f(x_i)~\Delta x = (-0.6+0.9+1.8)(2) = 4.2$
(b) We can use the left endpoint of each subinterval.
$x_1 = 3$
$x_2 = 5$
$x_3 =7$
$\sum_{i=1}^{3} f(x_i)~\Delta x = (-3.4-0.6+0.9)(2) = -6.2$
(c) We can use the midpoint of each subinterval.
$x_1 = 4$
$x_2 = 6$
$x_3 =8$
$\sum_{i=1}^{3} f(x_i)~\Delta x = (-2.1+0.3+1.4)(2) = -0.8$
Since the function is increasing, an estimate using the right endpoint of each subinterval is greater than the exact value of the integral.
Since the function is increasing, an estimate using the left endpoint of each subinterval is less than the exact value of the integral.
We can not know whether an estimate using the midpoint of each subinterval is less than, greater than, or exactly equal to the exact value of the integral.
