When approximating the area under a curve using left, right, or midpoint rectangles, the more rectangles you use, the better the approximation. So, “all” you’d have to do to get the exact area under a curve is use an infinite number of rectangles. Now, you can’t really do that, but with the fantastic invention of limits, this is sort of what happens. Here’s the definition of the definite integral that’s used to compute exact areas.
The definite integral(“simple“ definition): The exact area under a curve between x = a and x = b is given by the definite integral, which is defined as the limit of a Riemann sum:
Is that a thing of beauty or what? Note that this summation (everything to the right of “lim”) is identical to the formula for n right rectangles, Rn:
The only difference is that you take the limit of that formula as the number of rectangles approaches infinity
This definition of the definite integral is the simple version based on the right rectangle formula. You will see the real-McCoy definition in a moment, but because all Riemann sums for a specific problem have the same limit — in other words, it doesn’t matter what type of rectangles you use — you might as well use the right-rectangle definition. It’s the least complicated and it’ll always suffice.
Here is the exact area under f(x) = x2+ 1 between x = 0 and x = 3:
This result is pretty amazing if you think about it. Using the limit process, you get an exact answer of 12 — sort of like 12.00000000 … to an infinite number of decimal places — for the area under the smooth, curving function f(x) = x2+ 1, based on the areas of flat-topped rectangles that run along the curve in a jagged, sawtooth fashion.
Finding the exact area of 12 by using the limit of a Riemann sum is a lot of work (remember, you first have to determine the formula for n right rectangles). This complicated method of integration is comparable to determining a derivative the hard way by using the formal definition that’s based on the difference quotient.
Because the limit of all Riemann sums is the same, the limits at infinity of n left rectangles and n midpoint rectangles — for f(x) = x2+ 1 between x = 0 and x = 3 — should give you the same result as the limit at infinity of n right rectangles. Here’s the left rectangle limit:
And here’s the midpoint rectangle limit:
If you’re somewhat incredulous that these limits actually give you the exact area under f(x) = x2+ 1 between 0 and 3, you’re not alone. After all, in these limits, as in all limit problems, the arrow-number
is only approached; it’s never actually reached. And on top of that, what would it mean to reach infinity? You can’t do it. And regardless of how many rectangles you have, you always have that jagged, sawtooth edge. So how can such a method give you the exact area?
Look at it this way. Take a look at the next two figures.
f (x) = x2 + 1.”/>
Six “left” rectangles approximate the area under f (x) = x2 + 1.
You can tell from these figures that the sum of the areas of left rectangles, regardless of their number, will always be an underestimate (this is the case for functions that are increasing over the span in question).
And from the following figure, you can see that the sum of the areas of right rectangles, regardless of how many you have, will always be an overestimate (for increasing functions).
So, because the limits at infinity of the underestimate and the overestimate are both equal to 12, that must be the exact area. (A similar argument works for decreasing functions.)
All Riemann sums for a given problem have the same limit. Not only are the limits at infinity of left, right, and midpoint rectangles the same for a given problem, the limit of any Riemann sum also gives you the same answer. You can have a series of rectangles with unequal widths; you can have a mix of left, right, and midpoint rectangles; or you can construct the rectangles so they touch the curve somewhere other than at their left or right upper corners or at the midpoints of their top sides. The only thing that matters is that, in the limit, the width of all the rectangles tends to zero (and from this it follows that the number of rectangles approaches infinity). This brings you to the following totally extreme, down-and-dirty integration mumbo jumbo that takes all these possibilities into account.
The definite integral(real-McCoy definition): The definite integral from
is the number to which all Riemann sums tend as the width of all rectangles tends to zero and as the number of rectangles approaches infinity:
is the width of the ith rectangle and ci is the x-coordinate of the point where the ith rectangle touches f (x). (That
simply guarantees that the width of all the rectangles approaches zero and that the number of rectangles approaches infinity.)
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