Welcome to an example
on how to approximate the area
under a curve using rectangles.
We're asked to approximate the area
under the curve graphed below
from x=1 to x=6 using a Right Endpoint
approximation with 5 subdivisions
or five subintervals.
So because we're using rectangles
to approximate the area,
we can say that the area is going
to be approximately equal to the sum
of f[c] sub I times delta x,
where f[c] would be the height of each rectangle
and delta x would be the width of each rectangle.
Let's begin by determining delta x
which is equal to b-a/n,
where for this example,
because we're looking for the area
from x=1 to x=6,
a equals 1, b equals 6,
and because we're using 5 subdivisions
n equals five.
Which means delta x
is equal to 6-1/5,
which is equal to 1.
So we're going to have 5 subdivisions
each with a width of one unit.
And let's go ahead and show the subintervals on the graph.
Again we're concerned about the graph
from x=1
to x=6 which is here,
and now we'll form the subintervals
with the width of 1.
So we have our first subinterval here,
second subinterval here,
third subinterval here,
and then finally we have our fourth
and fifth subintervals here.
Next we'll form the rectangles
which we'll use to approximate the area,
and notice for this example,
we're using Right Endpoint approximations
which means we'll use the right side
of each subinterval to determine
the height of each rectangle.
So looking at the first subinterval from 1 to 2,
we're going to use the function value
of x=2 for the height
and therefore this would be the height
of the first rectangle.
Again because this is the function value
on the right side of this first subinterval.
Looking at the next subinterval,
notice how x=3 is on the right side of this subinterval
so we'll use the function value
of x=3, which is here,
for the height of the second rectangle.
For the next subinterval we'll use the
height of x=4, which would be here,
so this is the third rectangle.
For the next subinterval we'll use the
function value of x=5, which is here,
for the height of the rectangle.
And then finally for the last subinterval,
we'll use the function value of x=6,
which would be here,
for the height of this last rectangle.
So we can say the area
is approximately equal to
the area of these five rectangles
but let's first indicate the area using function notation,
and then because we're not given
the equation for the function,
we'll have to use the graph to
approximate the function values.
So the are that first rectangle is going to be,
again, the height times the width,
well the height is f[2], so f[2].
The width is delta x, which we know is always 1,
so f[2][1]
gives us the area of the first rectangle
+, second rectangle, notice how the height would be f[3],
so we'd do f[3], the width of [1]+,
next rectangle has a height of f[4],
so the area would be f[4][1]+,
the fourth rectangle, again using the right side,
the height would be f[5],
so f[5][1], for the area,
and then finally for the fifth rectangle,
the height would be f[6],
and therefore the area's f[6][1].
And now I'll approximate these
function values using the graph.
So notice where f[2],
we'll say the function value's approximately 2.1,
so the area's 2.1[1]+,
for the next area, we'd have f[3],
which looks like it's about 2.6,
so we'd have a area of 2.6[1]+,
f[4][1], f[4] is equal to 3,
so the area would be 3[1]+,
the next rectangle would have a height of f[5]
which looks like it's approximately 3.3.
So we'd have an area of 3.3[1]+,
for our last rectangle, the height is f[6],
f[6] looks like it's approximately 3.7,
so the area is 3.7[1].
'Course we could leave off multiplying by one,
but delta x is not always one,
so I'm leaving it in for consistency.
And now we'll go ahead and just find
some of these products
So we have 2.1
+2.6
+3
+3.3
+3.7.
So the area approximation using the Right Endpoints
is approximately 14.7 sq units.
Let's go ahead and shade the area of the five rectangles.
Notice how the area of each rectangle
is more than the area under the curve
in each subinterval,
and therefore this is often called
the upper sum approximation.
And it is important to recognize
that it's not the upper sum just
because we're using the Right Endpoints.
If this function was decreasing over the interval,
the Right Endpoints would actually give us the lower sum.
But it is true if the function is
increasing over an entire interval
then the Right Endpoints would be the upper sum.
We can also say that we have
five circumscribed rectangles.
If the rectangles were underneath the function
we could say that we had five inscribed rectangles.
I hope you found this helpful.