How do you use Riemann sums to evaluate the area under the curve of #1/x# on the closed interval [0,2], with n=4 rectangles using midpoint?

Question

Ezekiel Alexander · Accepted Answer

I will use what I think is the usual notation throughout this solution.

#int_0^2 1/xdx# Note that #f(x) = 1/x# and #a=0# and #b=2# #n=4# So #Deltax = (b-a)/n = (2-0)/4 =1/2# All endpoints: star with #a# and add #Deltax# successively: #0# #underbrace(color(white)"XX")_(+1/2)# #1/2# #underbrace(color(white)"XX")_(+1/2)# #1# #underbrace(color(white)"XX")_(+1/2)# #3/2# #underbrace(color(white)"XX")_(+1/2)# #2# All endpoints: #0#, #1/2#, #1#, #3/1#, #2# The subintervals are: #(0,1/2)#, #(1/2,1)#, #(1, 3/2)#, #(3/2,2)# We have been aske to use the midpoint of each subinterval as its sample point. The midpoints may be found by averaging the endpoints of each subinterval or by averaging the endpoints of the first subinterval to find its midpoint and then successively adding #Delta x# to get the others. The midpoints are: #1/4#, #3/4#, #5/4#, #7/4# Now the Riemann sum is the sum of the area of the 4 rectangles. We find the area of each rectangle by #"height" xx "base" = f("sample point") xx Deltax# Here we are using midpoints for sample points. So #R = f(1/4)*1/2+f(3/4)*1/2+f(5/4)*1/2+f(7/4)*1/2# #= (f(1/4)+f(3/4)+f(5/4)+f(7/4))1/2# Since #f(x)# is the reciprocal of #x#, we have #=(4/1+4/3+4/5+4/7)*1/2# Finish the arithmetic to finish.

Paisley Johnson · Answer

undefined To use Riemann sums to evaluate the area under the curve of $1/x$ on the closed interval $[0, 2]$ with $n = 4$ rectangles using the midpoint method, follow these steps:

1. Divide the interval $[0, 2]$ into $n$ subintervals of equal width. Since $n = 4$, each subinterval will have a width of $2/4 = 0.5$.
2. Determine the midpoints of each subinterval. For $n = 4$, the midpoints will be at $x = 0.25$, $x = 0.75$, $x = 1.25$, and $x = 1.75$.
3. Evaluate the function $1/x$ at each midpoint to get the corresponding function values.
4. Multiply each function value by the width of the subinterval ($0.5$ in this case\)).
5. Sum up all these products to get the Riemann sum.

Mathematically, the Riemann sum $S$ for $n$ rectangles using the midpoint method is given by:

$$
S = \sum_{i=1}^{n} f\left(\frac{x_{i-1} + x_i}{2}\right) \cdot \Delta x
$$

Where:
- $f(x)$ is the function $1/x$.
- $x_{i-1}$ and $x_i$ are the endpoints of the $i$th subinterval.
- $\Delta x$ is the width of each subinterval.

Substituting the values for the midpoint method with $n = 4$ and evaluating at the midpoints:

$$
S = \left( f(0.125) + f(0.375) + f(0.625) + f(0.875) \right) \cdot 0.5
$$

Then, calculate the function values at the midpoints and perform the summation to find the Riemann sum.