### Video Transcript

Calculate the right endpoint
estimate of the definite integral between zero and four of ๐ฅ squared plus two d๐ฅ
with ๐ 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 ๐ฅ-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 whatโs going on. Letโs begin by sketching the graph
of our function out: ๐ of ๐ฅ equals ๐ฅ squared plus two.

We have the curve represented by ๐
of ๐ฅ equals ๐ฅ squared plus two. And the area weโre interested in is
bounded by the vertical lines ๐ฅ equals zero and ๐ฅ 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 weโre going to do, though, is
split this region into two subintervals. And weโre 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, itโs probably fairly obvious
how weโre going to split this region up into two subintervals. But we can recall a formula, the
width of each subinterval given by ฮ๐ฅ is equal to ๐ minus ๐ over ๐, where ๐ and
๐ are the vertical lines that bound our region and ๐ 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 weโre
going to construct an extra vertical line at the point where ๐ฅ 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 ๐ of two. And the height of the second
rectangle will be the value of the function when ๐ฅ is equal to four. And of course, we know ๐ of ๐ฅ to
be equal to ๐ฅ squared plus two. So, ๐ of two is two squared plus
two, which is six. Similarly, ๐ of four is four
squared plus two, which is 18.

Remember, weโre 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. Thatโs simply the height of each
rectangle multiplied by its width, which we know to be two, which is equal to
48.

So weโve calculated an estimate for
the definite integral between zero and four of ๐ฅ squared plus two d๐ฅ. But is this an overestimate or an
underestimate? Well, if we simply go back to the
diagram, we can see that weโve actually calculated a little bit more area than we
need. And so, we can say that 48 is an
overestimate of the actual value.