Approximate the area under the curve over the specified interval by using the indicated number of subintervals (or rectangles) and evaluatin

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

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Cora 2 weeks 2021-09-27T14:05:44+00:00 1 Answer 0

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    2021-09-27T14:07:08+00:00

    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}

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45:7+7-4:2-5:5*4+35:2 =? ( )