Approximate the area under the curve over the specified interval by using the indicated number of subintervals (or rectangles) and evaluating the function at the left-hand endpoints of the subintervals. Compute the approximate area using up to 4 decimal places as needed. from to subintervals.

A) 11.625 square units
B) 23.25 square units
C) 19.25 square units
D) 46.5 square units
E) 2 square units

Question

Approximate the area under the curve over the specified interval by using the indicated number of subintervals (or rectangles) and evaluating the function at the left-hand endpoints of the subintervals. Compute the approximate area using up to 4 decimal places as needed. ​  from  to   subintervals.
​
A) 11.625 square units 
B) 23.25 square units 
C) 19.25 square units 
D) 46.5 square units 
E) 2 square units

Quizplus · Accepted Answer

We will use the Left Hand Rule to evaluate the integral.

The width of each rectangle is:

width = (b-a)/n = (4-2)/8 = 0.25

The height of each rectangle is the function evaluated at the left-hand endpoint of each subinterval.

f(2) = 1
f(2.25) = 1.35
f(2.5) = 1.6
f(2.75) = 1.85
f(3) = 2
f(3.25) = 2.15
f(3.5) = 2.2
f(3.75) = 2.25

The area of each rectangle is:

Area = (width)(height)

So the total area is:

Area ≈ 0.25(1) + 0.25(1.35) + 0.25(1.6) + 0.25(1.85) + 0.25(2) + 0.25(2.15) + 0.25(2.2) + 0.25(2.25) ≈ 23.25 square units.

Therefore, the best choice is answer B.

Approximate the Area Under the Curve Over the Specified Interval