This was the question I was given:
Use a Riemann sum with n = 5 rectangles to approximate the area of the region bounded by the lines $x = 1$, $x = 2$, $y = 0$ and the curve $y = 1/x$. Use the appropriate endpoint of each subinterval to compute a lower sum.
When I saw this problem, this is what I came up with: $$\sum^n_{i=0} \frac51\frac1x \Delta x$$
However, this did not result in the right answer. Where did I go wrong when finding the Riemann Sum? How can I rectify this?
Best Answer
I prefer to discuss Riemann sums in pictures. Here's the region whose area we are supposed to approximate:
You need to use $5$ rectangles. This means that you should subdivide the domain into $5$ equal pieces:
Next, we draw perpendicular lines up to the graph, ready to become the sides of rectangles:
Now, we we need to choose the height of the rectangles. We want the lower sums, hence we want the height of the rectangles to be as small as possible. As this is a decreasing function, the height of the rectangle will therefore be the function value at the rightmost point of its base, giving us our final picture:
These are our final five rectangles. If we compute the area of these rectangles, and sum them up, this will be our Riemann sum. Our rectangles all have a width of $0.2$. Their respective heights are $\frac{1}{1.2}, \frac{1}{1.4}, \frac{1}{1.6}, \frac{1}{1.8},$ and $\frac{1}{2}$. Thus, the Riemann sum is: