Source: http://www.dummies.com/how-to/education-languages/math/Calculus/Finding-Area.html
How to Approximate Area with Left Sums
You can approximate the area under a curve by using left sums. For example, say you want the exact area under
a curve between two points, 0 and 3. The shaded area on the left graph in the below figure shows the area you
want to find.
First, you get a rough
estimate of the area by
drawing three rectangles
under the curve, as shown
in the right graph in the
figure, and then
determining the sum of
their areas.
The rectangles in the figure represent a so-called left sum because the upper-left corner of each rectangle
touches the curve. Each rectangle has a width of 1 and the height of each is given by the height of the function
at the rectangle’s left edge. So, rectangle number 1 has a height of
its area (length x width or height x width) is thus 1 x 1, or 1. Rectangle 2 has a height of
so its area is 2 x 1, or 2. And rectangle 3 has a height of
so its area is 5 x 1, or 5. Adding these three areas gives you a total of 1 + 2 + 5, or 8. You can see that this is an
underestimate of the total area under the curve because of the three gaps between the rectangles and the curve
shown in the above figure.
For a better estimate, double the number of rectangles to six. The following figure shows six “left” rectangles
under the curve and also how the six rectangles begin to fill up the three gaps you see in the first figure.
See the three small shaded rectangles in the graph on the right in the second figure? They sit on top of the three
rectangles from the first figure, and they represent how much the area estimate has improved by using six
rectangles instead of three.
Now total up the areas of the six rectangles. Each has a width of 0.5 and the heights are f(0), f(0.5), f(1), f(1.5),
and so on. Here’s the total: 0.5 + 0.625 + 1 + 1.625 + 2.5 + 3.625 = 9.875. This is a better estimate, but it’s still
an underestimate because of the six small gaps you can see on the left graph in the above figure.
Here’s the fancy-pants formula for a left rectangle sum. The Left Rectangle Rule: You can approximate the
exact area under a curve between a and b,
with the sum of left rectangles given by the following formula. In general, the more rectangles, the better the
estimate.
Where n is the number of rectangles,
is the width of each rectangle, and the function values are the heights of the rectangles.
Look back to the six rectangles shown in the second figure. The width of each rectangle equals the length of the
total span from 0 to 3 (which of course is 3 – 0, or 3) divided by the number of rectangles, 6. That’s what the
does in the formula.
Now, what about those xs with the subscripts? The x-coordinate of the left edge of rectangle 1 in the second
figure is called
the right edge of rectangle 1 (which is the same as the left edge of rectangle 2) is at
right edge of rectangle 2 is at
the
the right edge of rectangle 3 is at
and so on all the way up to the right edge of rectangle 6, which is at
For the six rectangles in the second figure,
The heights of the six left rectangles in the second figure occur at their left edges, which are at 0, 0.5, 1, 1.5, 2,
and 2.5 — that’s
You don’t use the right edge of the last rectangle,
formula stops at
in a left sum. That’s why the list of function values in the
Here’s how to use the formula for the six rectangles in the second figure:
So the area under the curve is 9.875.
How to Approximate Area with Right Sums
You can approximate the area under a curve by using right sums. This method works just like the left sum
method except that each rectangle is drawn so that its right upper corner touches the curve instead of its left
upper corner.
For example, estimate the area under
from 0 to 3 by using right rectangles, as shown in this figure.
The heights of the three rectangles are given by the function values at their right edges: f(1) = 2, f(2) = 5, and
f(3) = 10. Each rectangle has a width of 1, so the areas are 2, 5, and 10, which total 17. This approximation
gives you an overestimate of the actual area under the curve.
The more rectangles you create between 0 and 3, the more accurate your estimate will be. Here's the Right
Rectangle Rule: You can approximate the exact area under a curve between a and b,
with a sum of right rectangles given by this formula:
Where, n is the number of rectangles,
heights of the rectangles.
is the width of each rectangle, and the function values are the
Now if you compare this formula to the one for a left rectangle sum, you get the complete picture about those
subscripts. The two formulas are the same except for one thing. Look at the sums of the function values in both
formulas. The right sum formula has one value,
that the left sum formula doesn’t have, and the left
sum formula has one value,
that the right sum formula doesn’t have. All the function values between
those two appear in both formulas. You can get a better handle on this by comparing the three left rectangles
from the below figure to the three right rectangles from the above figure.
For the left sum, you can find the areas and totals using the following formula:
So, three left rectangles add up to: 1 + 2 + 5 = 8
For the right sum, you can use the following formula:
So, three right rectangles add up to: 2 + 5 + 10 = 17
The sums of the areas are the same except for the left-most left rectangle and the right-most right rectangle.
Both sums include the rectangles with areas 2 and 5. If you look at how the rectangles are constructed, you can
see that the second and third rectangles in the second figure are the same as the first and second rectangles in
the first figure.
One last thing on this. The difference between the right rectangle total area (17) and the left rectangle total area
(8) — that’s 17 minus 8, or 9, in case you love calculus but don’t have the basic subtraction thing down yet —
comes from the difference between the areas of the two “end” rectangles just discussed (10 minus 1 is also 9).
All the other rectangles are a wash, no matter how many rectangles you have.
How to Approximate Area with Midpoint Rectangles
A good way to approximate areas with rectangles is to make each rectangle cross the curve at the midpoint of
that rectangle's top side. A midpoint sum is a much better estimate of area than either a left-rectangle or rightrectangle sum. The below figure shows why.
You can see in the figure that the part
of each rectangle that’s above the curve
looks about the same size as the gap
between the rectangle and the curve. A
midpoint sum produces such a good
estimate because these two errors
roughly cancel out each other.
The figure above shows how you’d use
three midpoint rectangles to estimate
the area under
from 0 to 3. For the three rectangles,
their widths are 1 and their heights are
f(0.5) = 1.25, f(1.5) = 3.25, and f(2.5) =
7.25. Area = base x height, so add 1.25
+ 3.25 + 7.25 to get the total area of
11.75.
Using the definite integral, you find that the exact area under this curve turns out to be 12, so the error with this
three-midpoint-rectangles estimate is 0.25. Contrast that with the much worse errors of the three-left-rectangles
estimate and the three-right-rectangles estimate of 4.0 and 5.0, respectively.
Here's the official midpoint rule:
Midpoint Rectangle Rule—You can approximate the exact area under a curve between a and b,
with a sum of midpoint rectangles given by the following formula. In general, the more rectangles, the better the
estimate:
Where, n is the number of rectangles,
is the width of each rectangle, and the function values are the heights of the rectangles.
How to Approximate Area with the Trapezoid Rule
With the trapezoid rule, instead of approximating area by using rectangles (as you do with the left, right, and
midpoint rectangle methods), you approximate area with — can you guess? — trapezoids.
Because of the way trapezoids hug the curve, they
give you a much better area estimate than either left
or right rectangles. And it turns out that a trapezoid
approximation is the average of the left rectangle and
right rectangle approximations. Can you see why?
(Hint: The area of each trapezoid is the average of the
areas of the two corresponding rectangles in the left
and right rectangle sums.)
The figure (left) shows three trapezoids drawn under
the function x2 + 1.
From the look of this figure, you might expect a
trapezoid approximation to be better than a midpoint
rectangle estimate, but in fact, as a general rule,
midpoint sums are about twice as good as trapezoid
estimates.
The Trapezoid Rule:
If you’ve already worked out the left and right
rectangle approximations for a particular function and
a certain number of rectangles, you can just average
them to get the corresponding trapezoid estimate (for
this problem, you know the answer you’re going to
get is (8 + 17)/2 = 12.5). If not, here’s the formula:
For the function in the above figure with three trapezoids, here’s the math:
Even though the formal definition of the definite
integral is based on the sum of an infinite number of
rectangles, you might want to think of integration as
the limit of the trapezoid rule at infinity. The further
you zoom in on a curve, the straighter it gets. When
you use a greater and greater number of trapezoids and
then zoom in on where the trapezoids touch the curve,
the tops of the trapezoids get closer and closer to the
curve. If you zoom in “infinitely,” the tops of the “infinitely many” trapezoids become the curve and, thus, the
sum of their areas gives you the exact area under the curve. This is a good way to think about why integration
produces the exact area — and it makes sense conceptually — but it’s not actually done this way.