How do you calculate the left and right Riemann sum for the given function over the interval [6,8], using n=2 for #f(x) = x^2#?

Question

Ezra Adams · Accepted Answer

See below.

Cut the interval #[6,8]# into #n=2# equal subintervals. The width (length) of each subinterval will be #Delta x = (b-a)/n = (8-6)/2 = 1# The subintervals are #[6,7]# and #[7,8]# For the left sum, use the left endpoints and add. #L_2 = f(6)*Deltax + f(7)*Delta x# # = 36+49 = 85# For the right sum, use the right endpoints and add. #R_2 = f(7)*Deltax + f(8)*Delta x# # = 49+64 = 113#

Paisley Johnson · Answer

undefined To calculate the left and right Riemann sums for the given function $ f(x) = x^2 $ over the interval $[6,8]$ using $ n = 2 $, we divide the interval $[6,8]$ into $ n $ subintervals of equal width.

With $ n = 2 $, the interval $[6,8]$ is divided into two subintervals:

1. $[6,7]$
2. $[7,8]$

For the left Riemann sum, we evaluate the function $ f(x) $ at the left endpoint of each subinterval and multiply it by the width of each subinterval.

For the right Riemann sum, we evaluate the function $ f(x) $ at the right endpoint of each subinterval and multiply it by the width of each subinterval.

Given that $ n = 2 $, we have:

1. Left Riemann sum:
$$ L_2 = f(6) \cdot \Delta x + f(7) \cdot \Delta x $$
where $ \Delta x = \frac{{b - a}}{n} = \frac{{8 - 6}}{2} = 1 $

2. Right Riemann sum:
$$ R_2 = f(7) \cdot \Delta x + f(8) \cdot \Delta x $$

Now, let's compute these values:

1. For the left Riemann sum:
$$ L_2 = f(6) \cdot 1 + f(7) \cdot 1 $$
$$ L_2 = 6^2 \cdot 1 + 7^2 \cdot 1 $$
$$ L_2 = 36 + 49 $$
$$ L_2 = 85 $$

2. For the right Riemann sum:
$$ R_2 = f(7) \cdot 1 + f(8) \cdot 1 $$
$$ R_2 = 7^2 \cdot 1 + 8^2 \cdot 1 $$
$$ R_2 = 49 + 64 $$
$$ R_2 = 113 $$

Therefore, the left Riemann sum $ L_2 $ for the given function over the interval $[6,8]$ using $ n = 2 $ is $ 85 $, and the right Riemann sum $ R_2 $ is $ 113 $.