We need to clarify what the title means. Here, “doing integrals” means computing specific
definite integrals, but instead of using traditional integration techniques like substitution,
by parts, or any antiderivative formulae, we would use geometry and physics to help
us out.
The reason why we can do integrals without integration is that definite integrals are
not defined in terms of antiderivatives, but as the area under the graph from lower to
upper limits. Of course, these two statements are equivalent and the equivalence is captured
in the Fundamental theorem of Calculus, so we can compute the area under the graph using
integration. However, as with many fields of Mathematics and problems in life, there
are not only 1 way to achieve the same task.
I know what you might be thinking: throughout this video, are we going to talk about how
to calculate the areas under these kinds of graphs? Well, of course not. Why would I dedicate
a video to explain how to calculate the area of rectangles, triangles and trapezium? All
the graphs considered in this video are curves, and we will be using Riemann sum and method
of exhaustion.
Although there is a time difference of 2 millennia between the discovery of the two methods,
their main ideas are very similar: to cut the curved region into a lot of smaller regions
with only straight lines as boundaries. The difference is that Riemann sum chop the curved
region into rectangles, and the other method chop it into triangles. To get the exact curved
region, we should have more and more of these regions. However, it is far easier to compute
the areas of these rectangular strips, so let's tackle the easier method first.
Here is a very classic example of using Riemann sums. This integral can be visually represented
like so. We are going to slice this region into several rectangular strips. Here, there
are n rectangular strips, and so the width of each rectangular strip is 1 over n. Now
the length of the rectangular strips is the y-coordinate of a point on the graph, which
is the function value at the corresponding value of x. In this case, the value of x is
k over n, where k denotes which slice you are working on. In this specific case, the
function is x squared, so the y-coordinate is k over n whole squared, which is the length
of the rectangular strip. Here, k ranges from 1 to n as we work from the 1st strip to the
nth one. Equipped with these two pieces of information, we can already compute the integral.
This integral is the sum of the areas of the rectangular strips, so we have this expression.
Of course, it is not exactly the area of the curved region, so we have to let n go to infinity.
We now just need to manipulate this algebraically by factoring out n^2 and bringing it to the
front, and we would have this nicer expression. (wait 1 or 2 sec) This bit here is known as
the sum of squares, and has this formula. There is a very nice animation to explain
the derivation of the formula done by a YouTube channel called Think Twice. Thanks for giving
permission to use it. Now with the formula proved, we go on with our algebraic manipulation.
We spread the 3 n’s with each factor in the numerator to get this. These expressions
would become 1, 0 and 0 respectively when n tends to infinity, and so replace these
with 1, 0, and 0, and now we get one-third. It’s important to see that no integration
is involved throughout this process. If you are a little bit confused, go ahead and pause
the video to think through everything we have gone through.
In general, if we want to compute the area under the graph from a to b, we first cut
the region up into n rectangular strips. Then the width of each rectangular strip is (b
- a) over n. Similar to the discussion a while ago, the length of the rectangular strip is
the function value at the endpoints of these rectangular strips. Here, k ranges from 0
to n-1, OR from 1 to n. Both are fine and we would use the one that is more convenient.
In this particular case, I have used the function value at the left of the rectangular strip,
so the range of k is from 0 to n-1, while in the case that we have done, we use the
function value at the right of the rectangular strip, so k ranges from 1 to n. The simple
reason why it doesn’t matter which range we are choosing is that we are going to let
n tend to infinity anyway, so the difference is negligible. Now we sum the areas of the
rectangular strips by multiplying the common width by the sum of the lengths. Remember
that there are two different ranges that k can take.
In reality, the exact Riemann sum definition is a lot more flexible, but this flexibility
does not make the computations more convenient. I will leave the exact definition up here
for anyone who wants to know these little details.
Now that we have used Riemann sums to compute the area under the graph of x^2, we would
continue to use it to compute the area under the graph of sin x in the next video. You
can also try it first before watching the next video.
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