Welcome to a lesson on using rectangles
to approximate the area under a curve
or under a graph.
If we can't find the exact area under a graph
using a geometric formula,
we can estimate the area under the graph using rectangles.
Let's consider the area under the graph of
f of x equals 2 divided by x
and above the x-axis on the closed interval
from one to five.
Notice how we don't have a geometric formula
that's going to give us this area here,
but we can approximate this area using rectangles.
To get an idea of how this works,
let's look at an animation.
What we can do is take the interval from one to five,
and divide it into equal sub-intervals.
In this case, each sub-interval
has a width of one unit,
and then we can form rectangles,
shown here to approximate the area under the curve.
Notice how for each rectangle,
we're using the function value on the left side
to determine the height of the rectangle.
And that's why these are called the left sided rectangles.
Also notice how the area of these four rectangles
would be more than the area under the curve,
which is shaded here in red.
And therefore,
this approximation would be called the upper sum.
Notice how as the number of rectangles increase,
our approximation gets better and better.
And just like here, we're using left sided rectangles.
We can also use right sided rectangles.
Shown here.
Notice when using right sided rectangles,
we use the function value
on the right side of each sub-interval
to determine the height of the rectangle.
Notice how here,
the area of the four rectangles would be
less than the area under the curve,
and therefore we'd call this approximation
the lower sum.
But once again,
notice how as the number of rectangles increase,
our approximation gets better and better.
Let's use four rectangles to approximate the area.
And we'll divide the closed interval from one to five
into four equal sub-intervals.
And we'll first use left-sided rectangles.
So n is equal to the number of rectangles.
So in this case, n is equal to four.
And because our interval is from one to five,
a equals one, b equals five.
So we'll first determine the width of each sub-interval,
or the width of each rectangle,
which is equal to delta x,
which would be b minus a divided by n,
which in this case is equal to one.
So each sub-interval is going to have a width of one,
and then because we're using left-sided rectangles,
the height of the rectangles will be equal
to the function value on the left of each sub-interval.
So our approximation for the area under the curve
is going to be the sum of the area of these four rectangles.
Let's call this first area A sub one,
followed by A sub two, A sub three,
and A sub four.
So our area approximation is going to be the sum
of these four areas.
And again, because we're using left-sided rectangles,
the area of A sub one is going to be equal to
f of x sub zero, which is f of one,
times delta x, which is equal to one.
The area of the next rectangle is going to be
f of x sub one, or f of two,
times delta x, or times one.
The area of the third rectangle is going to be equal to
f of x sub two, or f of three,
times delta x, or times one.
And finally, the area of the fourth rectangle
is going to be f of x sub three or f of four,
times delta x, or times one.
So we see that information here in these first two formulas.
So now we need to find these function values.
Find their products and the find the sum.
Well f of one is going to be two divided by one or two,
so this first area is two times one.
f of two is going to be two divided by two or one,
so the second area is going to be one times one.
This third area is going to be f of three,
which would be 2/3 times delta x,
which is one.
And finally, the fourth area would be f of four,
which would be two divided by four,
1/2 times delta x or times one.
And notice how the sum of these products
is equal to 25/6, which is approximately 4.167.
So this area approximation
would be called the upper sum,
and again, because the area of these four rectangles
is more than the area under the curve.
Now I do want to mention that
this is called the left-hand sum.
The left-hand sum is not always the upper sum,
it's only the upper sum if our function is decreasing
over the interval.
If our function was increasing,
the left-hand sum would actually be the lower sum.
Now let's do this again,
and this time we'll use right sided rectangles.
So again, the width of each rectangle
is still going to be equal to one.
But now the height of each rectangle will be determined by
the right side of each sub-interval,
rather than the left.
Notice how this first rectangle has a height of f of two,
not f of one, like we had when
we were using left sided rectangles.
So let's go through this process again.
We'll call this area sub one, area sub two, area sub three,
and area sub four.
So notice how for this first rectangle,
the height is going to be f of x sub one,
which is f of two.
And the width is going to be delta x,
which is equal to one.
So this first area is f of x sub one times delta x,
which is really f of two times delta x.
The second rectangle has a height
of f of x sub two, or f of three,
and a width of delta x or one.
So we have the second area as f of x sub two times delta x,
which is f of three times delta x.
The third rectangle has a height of f of x sub three,
which is f of four,
and a width of delta x or one.
So the area is f of x sub three times delta x,
or f of four times delta x.
And finally, for our last area of this rectangle
has a height of f of x sub four or f of five,
and a width of delta x or one.
Which again, is shown here in our formula.
So now we find these function values.
Find the products, and then find the sum.
Notice all the delta xs are equal to one.
f of two is equal to one.
f of three is equal to 2/3.
f of four is equal to 2/4 or 1/2.
And finally f of five is equal to 2/5.
So our area approximation now is 77/30
which is approximately 2.567.
And notice how here the area is less
than the area under the curve,
and therefore this would be considered the lower sum.
Let's look at left-sided and right-sided rectangles
next to each other.
So here are the left-sided rectangles
and here are the right-sided rectangles.
Notice for the left-sided rectangles,
we used the function values
that x equals one, two, three, and four,
to determine the height of each rectangle.
And for the right-sided rectangles,
we used the function values
that x equals two, three, four, and five
to determine the heights of the rectangles.
And of course if we increase the number of rectangles,
our approximation would be better,
so in general, we can say that if
f is continuous and nonnegative
on the closed interval from a to b,
then the area under the curve is going to
be approximately equal to the sum of
f of c sub i times delta x,
where these products represent the area of rectangles.
And notice how for our two examples
we used the left side and the right side
to determine the height of the rectangles.
In this formula here we're using c sub i,
where c sub i can be actually any value
in each sub-interval.
So we could use the midpoint
or any x value in the sub-interval
to make our approximation.
But the three most common x values to use
would be the left end point, the right end point,
or the midpoint of each sub-interval.
Let's end with a definition of
what the area would actually be equal to,
not just an approximation.
Let f be continuous and nonnegative
on the closed interval from a to b.
The area bounded by the graph of f and the x-axis,
and the vertical lines x equals a and x equals b
is equal to the limit as n approaches infinity
of the sum of f of c sub i times delta x
from i equals one to n.
So notice here, n is the number of rectangles,
so as n approaches infinity,
this limit is equal to the actual area under the curve
and above the x-axis on the interval from a to b.
Going back to our animation, this should make sense.
This is telling us that as
the number of rectangles approaches infinity,
this area is going to approach the area under the curve.
Again, it doesn't matter whether
you use left-side rectangles or right-sided rectangles,
the definition holds true.
In fact, we can use any x value in each sub-interval.
I hope you found this helpful.