Let 𝑓 of 𝑥 be equal to five over four 𝑥 for the interval 𝑥 between one and two. Using four subintervals and taking midpoints as the sample points, evaluate the Riemann sum of 𝑓 to six decimal places.
Before we go forward, let’s sketch this function on its interval. We know that we’re dealing with positive 𝑥-values because our interval is between one and two. Here is the interval for 𝑥 from one to two. If we plug in one for 𝑥, we see that 𝑦 equals 1.25. If we plug in two for 𝑥, 𝑦 equals 0.625. To give ourselves one more point to sketch the graph, we can calculate 𝑥 equals 1.5, which is 0.83 repeating. With this data, we can sketch the graph. We have something that would look like this.
Our function is bounded between one and two. And we want to divide this into four subintervals. We know that our subintervals must be of equal length. If we put a line at one and a half, we have two subintervals with the width of one-half. We can divide these two intervals and half again, like this. And we see that these subintervals all have a width of one-fourth. We’ll label our subintervals one through four.
In this problem, we want to find the Riemann sum, which is an approximation of an integral. And for us, that’s going to be an approximation of the area under this curve, all of this space. To do that, we need to turn these four subintervals into rectangles. We know the width of all four of our rectangles will be one-fourth. And we use the midpoint of each of these widths to find the height.
We know that halfway between one and one and a half is 1.25. Halfway between one and 1.25 is 1.125. Halfway between 1.25 and one and a half is 1.375. Halfway between 1.5 and two is 1.75. The midpoint of 1.5 and 1.75 is 1.625. And halfway between 1.75 and two is 1.875. So the midpoint of our first rectangle is 1.125, our second midpoint 1.375, our third midpoint 1.625, and our fourth midpoint 1.875.
These midpoints are important because we’ll calculate the value of our function at the midpoint. And that will become the height of our rectangle from that subinterval. 𝑓 of 𝑚 would equal five over four times 𝑚. For our first rectangle, the midpoint is 1.125. We plug that into the function. And we see the value of this function at 1.125 is ten-ninths. Our first interval, our first rectangle has a height of ten-ninths.
Now we need to find the area of this first rectangle. We do that by multiplying its width times its height, one-fourth times ten-ninths, to get the area of this space, which is five eighteenths.
We repeat this process for the remaining three rectangles. We plug in the midpoint of our second rectangle, five over four times 1.375, gives us a height of ten elevenths. We take our width of one-fourth and our height of ten elevenths and multiply them together to give us the area, which is five twenty-seconds.
Plug in the midpoint of our third rectangle, five divided by four times 1.625, gives a height of ten thirteenths. From there, we’ll multiply the width by the height. The area of the third rectangle is five twenty-sixths.
Calculate the function at the fourth midpoint, five over four times 1.875, gives a height of two-thirds. We multiply the width times the height to find the area, which is one-sixth. As the name sum implies, we need to add all four of these areas together. We take these four fractions and add them together using a calculator. It gives us 0.864024864. We want to round to six decimal places. There’s a four in the sixth decimal place. We look to the right of that four to see how we should round. Eight is greater than five. We should round up to 0.864025.