Answer
$-64~~$ is a lower estimate for the integral $\int_{10}^{30}f(x)~dx$
$16~~$ is an upper estimate for the integral $\int_{10}^{30}f(x)~dx$
Work Step by Step
Using the information in the table, we can use five subintervals estimate the value of the integral.
Since the function is increasing, to find a lower estimate of the integral, we can use the left-most value of each subinterval.
$x_1 = 10$
$x_2 = 14$
$x_3 = 18$
$x_4 =22$
$x_5 = 26$
$\sum_{i=1}^{5} f(x_i)~\Delta x = (-12-6-2+1+3)(4) = -64$
To find an upper estimate of the integral, we can use the right-most value of each subinterval.
$x_1 = 14$
$x_2 = 18$
$x_3 =22$
$x_4 = 26$
$x_5 = 30$
$\sum_{i=1}^{5} f(x_i)~\Delta x = (-6-2+1+3+8)(4) = 16$
$-64~~$ is a lower estimate for the integral $\int_{10}^{30}f(x)~dx$
$16~~$ is an upper estimate for the integral $\int_{10}^{30}f(x)~dx$
