Evaluate the Riemann sum for f(x) = 3x − 1, −6 ≤ x ≤ 4, with five subintervals, taking the sample points to be right endpoints.

Question

Evaluate the Riemann sum for f(x) = 3x − 1, −6 ≤ x ≤ 4, with five subintervals, taking the sample points to be right endpoints.

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Clara 2 days 2021-10-12T13:07:56+00:00 1 Answer 0

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    2021-10-12T13:09:13+00:00

    Answer:

    The Riemann sum equals -10.

    Step-by-step explanation:

    The right Riemann Sum uses the right endpoints of a sub-interval:

    \int_{a}^{b}f(x)dx\approx\Delta{x}\left(f(x_1)+f(x_2)+f(x_3)+...+f(x_{n-1})+f(x_{n})\right)

    where

    \Delta{x}=\frac{b-a}{n}

    To find the Riemann sum for \int\limits^{4}_{-6} {3x-1} \, dx with n = 5 rectangles, using right endpoints you must:

    We know that a = -6, b = 4 and n = 5, so

    \Delta{x}=\frac{4-\left(-6\right)}{5}=2

    We need to divide the interval −6 ≤ x ≤ 4 into n = 5 sub-intervals of length \Delta{x}=2

    a=\left[-6, -4\right], \left[-4, -2\right], \left[-2, 0\right], \left[0, 2\right], \left[2, 4\right]=b

    Now, we just evaluate the function at the right endpoints:

    f\left(x_{1}\right)=f\left(-4\right)=-13=-13

    f\left(x_{2}\right)=f\left(-2\right)=-7=-7

    f\left(x_{3}\right)=f\left(0\right)=-1=-1

    f\left(x_{4}\right)=f\left(2\right)=5=5

    f\left(x_{5}\right)=f(b)=f\left(4\right)=11=11

    Finally, just sum up the above values and multiply by 2

    2(-13-7-1+5+11)=-10

    The Riemann sum equals -10

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