How do you use Riemann sums to evaluate the area under the curve of #f(x)=x^3# on the closed interval [1,3], with n=4 rectangles using right, left, and midpoints?
1 Answer
I will use what I think is the usual notation throughout this solution.
Note that
All endpoints: star with
The subintervals are
Now the Riemann sum is the sum of the area of the 4 rectangles. We find the area of each rectangle by
For the left sum,
Left Endpoints:
So,
#= (f(1)+f(3/2)+f(2)+f(5/2))1/2#
# = (1+27/8+8+125/8)1/2 = 14#
For the right sum, use the right endpoints of the subintervals
Right Endpoints:
I'll leave the arithmetic to the reader.
For midpoints, we take the midpoint of each subinterval.
The subintervals are
The midpoint is the average of the endpoints. Add the endpoints and divide by 2, (Or find the first one and successively add
Midpoints:
The arithmetic is left to the reader.