Q:

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

Accepted Solution

A:
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