Use the given information to find the left- and right-hand Riemann sums for the following function. If necessary, round your answers to five decimal places.

Question

²

, , and 

Left-Hand Riemann Sum  = _____

Right-Hand Riemann Sum = _____

Answer 1

integral of x^2 +3 = x^3/3 + 3x

evaluated from -3 to 5

= (5^3)/3 + 3(5) - [(-3)^3/3 + 3(-3)]

= 125/3 +15 +9 +9

= 125/3 + 33

= 125/3 + 99/3 = 224/3 = 74 2/3

when you do the Reiman sums, you should get near 74 2/3

with n=16, the base of each rectangle = (5+3)/16 = 1/2

the height of each of 16 rectangles from the left = f(-3), f(-2.5), f(-2), f(-1.5), f(-1), f(-.5), f(0), f(.5), f(1), f(1.5), f(2), f(2.5), f(3), f(3.5), f(4), f(4.5)

= 12, 9.25, 7, 5.25, 4, 3.25, 3, 3.25, 4, 5.25, 9.25, 12, 15.25, 19, 23.25

add them up: 2(28.25 +12.5) + 15.25 + 42.25 = 2(40.75) + 57.5 = 81.5+57.5 = 138

multiply by the rectangles' base 1/2 = 138(1/2) = 69.00000

for the right = same calculations, except replace f(-3) with f(5)=28, 28-12 = 16.

add 16 to the sum of heights =138+16 = 152

152/2 = 76.00000

Left sum approximation = exactly 69

Right sum approximation = exactly 76

exact area = 74 2/3