Find an approximation of the area of the region r under the graph of the function f on the interval [-1, 2]. use n = 6 subintervals. choose the representative points to be the left endpoints of the subintervals. f(x) = 7 - x2

The approximation of the area of the region R under the graph of the function is 18.625 square units.

In this question we must calculate the approximate area () below the curve by means of Riemann sums with left endpoints, whose expression is described below:

, for (1)

(2)

Where:

- Lower bound - Upper bound - Number of subintervals - Index

If we know that , , and , then the approximate area is:

The approximation of the area of the region R under the graph of the function is 18.625 square units.

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What is the width of each interval?  To answer this, subtract -1 from 2.  Result:  3.  If n=6, then the width of each interval is 3/6, or 1/2.

If we're to use the left endpoints, then the x values in question are {-1, -1/2, 0, 1/2, 1, 3/2, 2}.

Evaluate the given function at these 6 x-values.  Answer set (to 2 decimal place accuracy):  {3,3.75, 4, 3.75, 3, 1.75).

Now calculate each of the 6 sub-areas under the curve.  To do this, multiply each of the 6 "heights" given immediately above by the subinterval width (1/2).  You will then have 6 subareas.  Find the sum of these 6 subareas.

Your result here is the approx. area under the curve of f(x)=7-x between x=-1 and x=2.

6(2x + 5y)6 (2x) = 12x6 (5y) = 30y12x + 30y