The table shows the number of people watching a sports TV channel, measured at various intervals during a day. Use a trapezoid approximation with six subintervals to estimate the total number of people who watched the channel during that day.
Let’s draw a graph to plot this information. As we were given the viewings in people per minute, let’s use minutes on the 𝑥-axis, starting with midnight as zero and the following midnight as 24 hours or 1440 minutes. We can then do the same for the rest of the times, calculating how many minutes after midnight each time is. And then, we can finish labeling the 𝑥-axis. We can now roughly plot the points from the table and then create a smooth curve. Now, the area under the curve will give an approximation for the total number of people who watch the sports channel in this 24-hour period.
One method we know to find the area between a curve and the 𝑥-axis is to integrate. Let’s say this curve is 𝑝 of 𝑡, a function of time. We could integrate this between zero and 1440 with respect to time. But we don’t actually know what this function is. So instead, we make an approximation using subintervals. And whilst we could use a series of rectangles to estimate our area, it’s much more accurate to use trapezoids. So, here we have our six trapezoids. Of course, we have the trapezoid rule, which gives us a nice formula to work this out. However, this only works when we have even subintervals. And these subintervals are clearly not all the same width. So, we’re going to calculate each area one by one.
Let’s firstly recall that, for any trapezoid with bases 𝑎 and 𝑏 and height ℎ, the area is 𝑎 add 𝑏 over two multiplied by ℎ. And this just takes an average of the two bases and multiplies it by the height. But when we’re using trapezoids to approximate the area under a curve, we need to think of the trapezoids as being rotated by 90 degrees, like this. Of course, this method isn’t going to give us the exact answer. It’s just an approximation to this integral.
So, let’s start with the first trapezoid. The two side lengths are three and one as we can see from the table. So, we add these together, divide by two, and then multiply by the width of this trapezoid, which we can see is 300. We can then move on to the next trapezoid. Its side lengths are one and two. So, we add these together and divide by two. We then multiply by the width of this trapezoid, which is 240. We can then move on to the next trapezoid. This has side lengths two and five and width 180. The fourth trapezoid has side lengths of five and 15 and a width of 240. The fifth trapezoid has side lengths of 15 and 10 and a width of 300. And finally, the last trapezoid has side lengths of 10 and three and a width of 180.
We can then calculate each of these areas to be 600, 360, 630, 2400, 3750, and 1170. We can then add them all together to give us 8910 as our estimate for the total number of people who watched the channel during this day.