Question

Use the midpoint rule with the given value of n to approximate the integral. (Round your answer to four decimal places.)

16
0
sin
x
dx, n = 4

Answer 1

Split up [0, 16] into 4 equally-spaced subintervals of length $\frac{16-0}4=4$,

[0, 16] = [0, 4] U [4, 8] U [8, 12] U [12, 16]

with midpoints 2, 6, 10, and 14, respectively.

Then with the midpoint rule, we approximate the integral to be about

$\displaystyle \int_0^{16} \sin(\sqrt x) \, dx \approx 4 \left(\sin(\sqrt2) + \sin(\sqrt6) + \sin(\sqrt{10}) + \sin(\sqrt{14})\right) \approx \boxed{4.1622}$