How do you use n = 5 equally divided subdivisions to estimate the integral from 1 to 3 of #(1/x^2)dx# with the right-hand Riemann sum?

Question

Christopher Agresta · Accepted Answer

See the explanation.

#f(x) = 1/x^2# The interval #[a,b]# is #[1,3]# and #n = 5# So #Delta x = (b-a)/n = (3-1)/5 = 2/5# The subintervals are: #[1, 7/5]#, #[7/5, 9/5]#, #[9/5, 11/5]#, #[11/5, 13/5]#, #[13/5, 3]# The right endpoints are: #7/5, 9/5, 11/5, 13/5, 3# Find #f# of each right endpoint times #2/5# (because #Delta x = 2/5# is the base of each rectangle.) and then add. (Or find #f# of each right endpoint, add those and then multiply by #2/5#) #f(7/5)*2/5+ f(9/5)*2/5 + f(11/5)*2/5 + f(13/5)*2/5+f(3)*2/5# (or #(f(7/5)+ f(9/5) + f(11/5) + f(13/5)+f(3))*2/5#

Emilia Aguilar · Answer

undefined To estimate the integral from 1 to 3 of $ \frac{1}{x^2} $ using the right-hand Riemann sum with $ n = 5 $ equally divided subdivisions, follow these steps:

1. Calculate the width of each subdivision, $ \Delta x $, by dividing the total interval length by the number of subdivisions:
$$ \Delta x = \frac{3 - 1}{5} = \frac{2}{5} $$

2. Determine the endpoints of each subdivision. Since we are using right-hand Riemann sum, the right endpoint of each subdivision will be used for evaluation.

3. Evaluate the function $ \frac{1}{x^2} $ at the right endpoint of each subdivision.

4. Multiply each function value by the width of the subdivision, $ \Delta x $.

5. Sum up all these products. This sum represents the estimate of the integral using the right-hand Riemann sum with $ n = 5 $ subdivisions.

6. The formula for the right-hand Riemann sum is:
$$ R_n = \sum_{i=1}^{n} f(x_i^*) \Delta x $$
where $ x_i^* $ is the right endpoint of the $ i $-th subdivision.