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 right-hand endpoints of the subintervals. (See Example 1.)
f(x) = 9 − x2 from x = 1 to x = 3; 4 subintervals

Answer 1

Split up [1, 3] into 4 subintervals:

[1, 3/2], [3/2, 2], [2, 5/2], [5/2, 3]

each with length (3 - 1)/2 = 1/2.

The right endpoints $r_i$ are {3/2, 2, 5/2, 3}, which we can index by the sequence

$r_i=1+\dfrac i2$

with $1\le i\le4$.

Evaluating the function at the right endpoints gives the sampling points $f(r_i)$, {27/4, 5, 11/4, 0}.

Then the area is approximated by

$\displaystyle\int_1^3f(x)\,\mathrm dx\approx\frac12\sum_{i=1}^4f(r_i)=\frac12\left(\frac{27}4+5+\frac{11}4+0\right)=\boxed{\frac{29}4}$