Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of f and the x-axis over the interval.

Question

Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of f and the x-axis over the interval.

FunctionInterval
f(x) = 3x2 + 8	[−1, 3]

 Find the exact area.

Answer 1

N=4, the subintervals are [-1,0] U [0,1] U [1,2] U [2,3]

the midpoints of them are x=-1/2, 1/2, 3/2, 5/2

f(-1/2) = 3(-1/2)^2 + 8 = 3/4+8 = 8.75

f(1/2) = 3(1/2)^2 + 8 = 8.75

f(3/2) = 3(3/2)^2 + 8 = 3(9/4) + 8 = 27/4 + 8 = 59/4 = 14.75

f(5/2) = 3(5/2)^2 + 8 = 3(25/4) + 8 = 75/4 + 32/4 = 107/4 = 26.75

the width of the rectangles are 1, so the areas are the sum of the

function values: 59

integrating:

(1/3)(3) x^3 + 8x

limit as x-->3 : (1/3)(3)(3^3) + 8*3 = 27 +24 = 51

limit as x-->-1: (1/3)(3)(-1)^3 + 8(-1) = -1 + -8 = -9

subtracting the limits: 51--9 = 60

not too bad!!!