Welcome to an example
of how to approximate area under a curve,
using midpoint approximation.
So here we're asked to approximate the area
under the curve graphed below
from X equals one to X equals six
using a midpoint approximation with five subdivisions.
So because we're using rectangles to approximate the area,
we can say the area is going to be
approximately equal to the sum of
F of C sub I times delta X,
where as C sub I is going to be
the height of each rectangle,
and delta X would be the width of each rectangle.
Let's being by determining delta X,
where delta X is always equal to B minus A divided by N.
Where in our case, because we're considering the graph,
from X equals to one to X equals six,
A equals one, and B equals six.
And because we're using five subdivisions,
N, the number of rectangles, is going to be five.
So we can say that delta X
is equal to B minus A divided by N
is going to be six minus one divided by,
which equals one.
Let's go ahead and form the subintervals on our graph.
Again, we're starting X equals one,
and again X equals six,
and now we'll form five subintervals,
each with the width of one.
So we have a subinterval here,
here, here, and here.
And now let's sketch the rectangles we'll use
to approximate the area under the curve.
And again we're asked to use a midpoint approximation,
we'll use the function value
at the midpoint of each subinterval
as the height of each rectangle.
So notice the first subinterval one to two,
the height of rectangle is going to be F of 1.5,
because 1.5 would be the midpoint between one and two.
So now the rectangle would be here,
and this would be the rectangle for the first subinterval.
The midpoint between two and three would be 2.5,
so the function value of 2.5 is going to give us
the height of the second rectangle, which would be here.
So this would the height of the second rectangle.
Next, the function value of X equals 3.5
would give us the height of the third rectangle,
which would be here,
so here's our third rectangle.
The fourth rectangle, the height is going to be F of 4.5
which would be here.
And finally for the last subinterval,
because 5.5 would be the midpoint between five and six.
F of 5.5 are here,
would be the height of this last rectangle.
So could say that the area is approximately...
Let's first write the area using function notation,
and then because we're not give the function
that gives us this blue graph,
we'll have to use the graph
to approximate the function values.
So the area this first rectangle
is going to be the height times the width,
which would be F of 1.5 times one.
And the second rectangle,
the area would be F of
2.5 times one.
Plus, with the third rectangle,
we have F of 3.5 times one for the area.
The fourth rectangle,
we have an area of F of 4.5 times one.
And then finally for the last rectangle,
we have an area of F of 5.5 times one.
Now, we'll approximate these function value using the graph.
So for F of 1.5, let's say the function value
is approximately 1.8, so the area is going to 1.8 times one.
Plus, for F of 2.5,
we'd approximate that function value, as a I say, 2.3.
Remember, function values are Y values.
We could leave off the times one,
but double X is not always one,
so I'll leave it here for consistency.
The next area is going to be F of 3.5 times one.
F of 3.5 is approximately 2.8.
So we have 2.8 times one, plus...
And the fourth rectangle, we have F of 4.5 times one.
F of 4.5 is approximately 3.2,
so we have 3.2 times one.
And then finally, the area of the last rectangle,
we have F of 5.5,
we shall say is 3.5 times one.
Let's go ahead and find the sum of these products.
We have 1.8, plus 2.3,
plus 2.8,
plus 3.2,
and finally plus 3.5.
Enter.
So our approximation of the area under the curve,
using the midpoint approximation,
would be approximately 13.6 square units.
Let's go and shade the area of the five rectangles.
Looking at the graph,
notice how it looks like this midpoint approximation
is a very good approximation for the area under the curve.
Notice how the rectangles are missing
a little bit of area on the set of each subinterval,
but they also include extra area
on the left side of each sub interval.
I hope you find this helpful.