Calculate the right endpoint
estimate of the definite integral between zero and four of 𝑥 squared plus two d𝑥
with 𝑛 equals two subintervals. Is the result an overestimate or
underestimate of the actual value?
The right endpoint estimate or the
right Riemann sum is a way of estimating a definite integral by thinking about the
area between the curve and the 𝑥-axis. In a right endpoint estimate, we
split that area into a given number of rectangles. The total area of those rectangles
gives us an estimate for the definite integral. So we can see what’s going on. Let’s begin by sketching the graph
of our function out: 𝑓 of 𝑥 equals 𝑥 squared plus two.
We have the curve represented by 𝑓
of 𝑥 equals 𝑥 squared plus two. And the area we’re interested in is
bounded by the vertical lines 𝑥 equals zero and 𝑥 equals four. Those are simply the upper and
lower limits of our definite integral. Now, if we were to calculate the
definite integral, it would tell us the exact area shaded. What we’re going to do, though, is
split this region into two subintervals. And we’re going to form two
rectangles. The height of each rectangle will
be the value of the function at the right endpoint of each subinterval.
Now, it’s probably fairly obvious
how we’re going to split this region up into two subintervals. But we can recall a formula, the
width of each subinterval given by Δ𝑥 is equal to 𝑏 minus 𝑎 over 𝑛, where 𝑎 and
𝑏 are the vertical lines that bound our region and 𝑛 is the number of
subintervals. In this case, the width of each
subinterval will be four minus zero over two which is simply two. We count up a multiples of two. And actually, we see that we’re
going to construct an extra vertical line at the point where 𝑥 equals two.
And so, we have our two
rectangles. The height of the first rectangle
will be the value of the function at the right endpoint, so 𝑓 of two. And the height of the second
rectangle will be the value of the function when 𝑥 is equal to four. And of course, we know 𝑓 of 𝑥 to
be equal to 𝑥 squared plus two. So, 𝑓 of two is two squared plus
two, which is six. Similarly, 𝑓 of four is four
squared plus two, which is 18.
Remember, we’re going to calculate
the area of each of these rectangles. Now, since the area of a rectangle
is simply its base multiplied by its height, we can say that an estimate to our
definite integral is six multiplied by two plus 18 multiplied by two. That’s simply the height of each
rectangle multiplied by its width, which we know to be two, which is equal to
So we’ve calculated an estimate for
the definite integral between zero and four of 𝑥 squared plus two d𝑥. But is this an overestimate or an
underestimate? Well, if we simply go back to the
diagram, we can see that we’ve actually calculated a little bit more area than we
need. And so, we can say that 48 is an
overestimate of the actual value.