Approximating area with right and left endpoints

Question

Estimate the area under the graph of  over the interval  using four approximating rectangles and right endpoints. Repeat the approximation using left endpoints.

Answer 1

it helps to draw a sketch of the graph

each rectangle has a base of 1/2

for right end points, if you mean f(x) = 1/(x+4) then

height of the rectangles are f(-2.5) = 1/1.5

f(-2) =1/2

f(-1.5) = 1/2.5

f(-1) = 1/3

sum them to get (1/2)[1/1.5 + 1/2 + 1/2.5 + 1/3]

= (1/2)(5/6 + (5+3)/7.5) = .5(5/6+ 8/7.5) = about .417 + 1.067/2 = .417+ .534 = .951

for left endpoints

replace f(-1) with f(-3)

to get (1/2)[1 +1/1.5 + 1/2 + 1/2.5] = (1/2)(1.5 + 8/7.5) =about .5(1.5 +1.067) = .5(2.567) = 1.284

average them gives (.951+1.284)/2 = 1.118

for f(x) = 1/(x+4)

integral = F(x) = ln(x+4) +C

evaluated from -3 to -1

= ln3 - 0 = about 1.099 = area under the hyperbolic curve which is in between the 2 approximations

rounded to one decimal place, it's the same as the average of right and left end point approximations