. (5 points)
Kim G. answered • 01/20/21
Yale student pursuing MA in statistics and PhD in public health
Hi Nic! This is a good question - it helps us understand what integrals fundamentally are. I'm new to Wyzant, and I'm still not sure how to share a drawing. So here's my best effort to explain in words what's happening:
Fundamentally, an integral is the area under a curve f(x) over a certain x interval. When we can't calculate an integral directly, it's helpful to use what we already know about calculating areas to approximate the integral.
It looks like we have a bunch of x values between 1 and 2, with corresponding values of f(x). We're asked first to divide the information we're given into five connected "rectangles", then use the area of the rectangles to estimate the area under the curve f(x) (i.e. the integral). We calculate the area of the rectangles using the right-most x-value of each rectangle as the height. Here's how I'd approach this:
1) Divide the table into five, equal length x intervals, which will become the "base" of each rectangle. Believe it or not, I think this is one of the hardest parts of the problem. We want to draw five connected rectangles using the values given in this table. To do this, my first impulse is to find rectangles of equal width that are connected. That means that they'll share some endpoints. Thankfully, I can see a pretty straightforward way to do that:
Rectangle 1 will go from 1 to 1.2
Rectangle 2 will go from 1.2 to 1.4 (note the shared endpoint with Rectangle 1)
Rectangle 3 will go from 1.4 to 1.6 (again note the shared endpoints with Rectangle 2, and so on)
Rectangle 4 will go from 1.6 to 1.8
Rectangle 5 will go from 1.8 to 2.0
Note that each interval is of width 0.2. This will be important when we're calculating the area of the rectangle: base times height.
2) Identify the "height" of each rectangle. This problem tells us to use the right-most x value to calculate the height of each rectangle. What are those? Let's take a look at the table we're given to find out.
Rectangle 1's right endpoint is 1.2: the corresponding "height" is 0.833
Rectangle 2's right endpoint is 1.4: the corresponding "height" is 0.714
Rectangle 3's right endpoint is 1.6: the corresponding "height" is 0.625
Rectangle 4's right endpoint is 1.8: the corresponding "height" is 0.556
Rectangle 5's right endpoint is 2.0: the corresponding "height" is 0.500.
3) For each rectangle, multiply base times height to get the area.
Rectangle 1: 0.2 x 0.833 = 0.1666
Rectangle 2: 0.2 x 0.714 = 0.1428
Rectangle 3: 0.2 x 0.625 = 0.125
Rectangle 4: 0.2 x 0.556 = 0.1112
Rectangle 5: 0.2 x 0.500 = 0.1
4) Add each of the values from step 3 to get our integral approximation. I usually use Excel spreadsheets for these types of calculations, but really you can use any calculator (or you can do it in your head if you're fancy). For my answer, I got 0.1666 + 0.1428 + 0.125 + 0.1112 + 0.1 = 0.6456.
There you have it! Hope this helps :)
