Welcome to an example
in which we'll approximate the area under a curve
using rectangles.
We're asked to approximate the area
under the curve graphed below
from x equals one to x equals six
using a left endpoint approximation
with five subdivisions or five subintervals.
To approximate this area, we'll be determining
a Riemann sum where the area is approximately equal to
the sum of f of c sub i
times delta x
where f of c sub i would be the height of the rectangle
and delta x would be the width.
Let's begin by determining delta x,
which will give us the width of each subinterval.
Delta x is equal to b minus a divided by n.
In this example, a equals one and b equals six,
which means delta x
is equal to six minus one divided by n
where n is the number of subdivisions or subintervals,
which is five.
So we have five divided by five, which equals one.
Let's go ahead and sketch the subintervals from one to six.
We're starting at one,
ending at six,
and each subinterval has a width of one,
so these would be the subintervals.
Now let's go ahead and sketch the five rectangles
that we'll use to approximate the area.
Because we're using a left endpoint approximation,
we'll use the left side of each subinterval
to determine the height of each rectangle.
So, looking at this first subinterval,
we're going to use the left side, or this side here,
to determine the height of the rectangle.
Therefore, this would be the height of the first rectangle
in the first subinterval.
We'll look at using the right side as well as the midpoint
in the next two videos.
Again, in this example, we're using the left side
or left endpoint to determine the height of each rectangle.
For the next subinterval,
this is the left side of the subinterval,
so this would be the height of the next rectangle
or second rectangle.
The third subinterval, we're going to use
this function value here as the height.
The next subinterval,
this would be the height of the rectangle.
Finally, for the last subinterval,
this would be the height of the last rectangle.
Notice how we're not given the function for this blue graph,
so we'll have to approximate the function values
using the graph.
Let's first write this out using function notation.
We would have the area
is approximately equal to
for the first rectangle, we'd have f of one
times delta x, which is one.
Plus for the second rectangle,
the height would be f of two.
The width, again, is one,
plus the third rectangle would have a height of f of three,
a width of one, plus the fourth rectangle
would have a height of f of four,
a width of one.
Finally, the fifth rectangle would have a height of
f of five
and a width of one.
Now we're going to approximate these function values
using the graph.
Remember, function values are y values.
So f of one is approximately, let's say, 1.5,
so we'd have 1.5 times one.
Of course, we could leave off the times one,
but it's not always one,
so we'll leave it in to be consistent.
Plus f of two looks like it's going to be
approximately, let's say, 2.1.
So we have 2.1 times one
plus f of three.
Looks like it's going to be approximately, let's say, 2.6.
Width of one, plus f of four
is equal to three,
so we have three times one, plus f of five.
Looks like it's going to be approximately 3.3.
So we'd have 3.3 times one.
Now, remember, we are approximating these function values,
so there is going to be a margin of error,
but as long as we're close,
our homework system will give us credit.
We have 1.5
plus 2.1
plus 2.6
plus three plus 3.3.
So the approximation of the area under the curve
using these five rectangles would be 12.5
square units.
Let's go ahead and shade the area of the rectangles.
Notice how the area of the rectangles
would be less than the area under the curve
over this interval.
Because of this, this is sometimes called the
lower sum or lower approximation.
Just because we're using the left endpoints
doesn't mean it's always going to be the lower sum.
It depends on whether the function
is increasing or decreasing over the given interval.
In this case, notice how the function
is increasing over this interval.
Because the function is increasing over this interval
and the function is not negative,
using the left endpoints does give us the lower sum
or lower approximation.
Meaning the actual area under the curve
would be more than 12.5 square units.
Again, next we'll take a look at using
the right endpoint followed by using the midpoints
in the next two videos.
I hope you found this helpful.