Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of the function and the x-axis over the given interval. (Round your answer to three decimal places.)

Question

f(x) = 4 tan x, [0, 𝜋/3]

PLEASE HELP ASAP

Answer 1

so the interval is broken up at x = 0, π/12, π/6, π/4, π/3
each interval has width π/12, and each region is a trapezoid, with area
(f(x) + f(x+π/12))/2 * π/12 for x = 0, π/12, π/6, π/4
So review the midpoint rule (or trapezoid rule) to understand the shortcut for the computation.
Post your work if you get stuck

Answer 2

i got 2.09, idk if it is right

Answer 3

Sounds like you're stuck Why did you not show your work?

Actually, I got that wrong. The midpoint rule is not the same as the trapezoidal rule. We are still using rectangles. So the calculation is

π/12 (f(π/24)+f(3π/24)+f(5π/24)+f(7π/24)) = 2.74
Not sure how you got 2.09

Since the graph is concave up. a Riemann sum will underestimate. In fact,
∫[0,π/3] 4tan(x) dx = 2.77

Answer 4

a handy calculator is at

www.emathhelp.net/en/calculators/calculus-2/midpoint-rule-calculator/?f=4tan%28x%29&a=0&b=pi%2F3&n=4