Perfect Approximations!

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We discuss how to approximate the area under a curve.

We’ve seen how we can approximate the area under a curve using a Riemann Approximation; leftend, rightend, or mid-point approximations. But none of these have really used our keystone calculus tool - the limit, what gives? Well, what if we want an exact value, and not just an approximation? This is where we will use our limiting process.

Video Lecture

Text with Additional Details

This question is remarkably close to one of the first questions we posed this semester: how do you determine the exact value of $\pi$ , rather than getting an approximation using the “method of exhaustion” from the Ancient Greeks? The answer then, is the answer now - we need limits!

To see how we use limits, watch the above video for a nice visualization.

1 : What about the Riemann Approximation method suggests that we want to use limits to get an exact answer?

It’s in a calculus class, so it’s going to involve limits sooner or later. The more rectangles we use, the better approximation we tend to get. Whenever we want to use/do something “infinitely many” times, that’s almost always a limit! We want to fill in a round shape with square boxes. Since the shapes aren’t the same, it must use limits. Because limits are annoying and so is math class. It was inevitable.

So, we’ve seen that by taking a limit of the number of rectangles to infinity, we can get a perfect approximation of the area under the given curve. The formula; $\displaystyle \lim \limits _{N\rightarrow \infty } \sum \limits _{k=1}^N \frac {b-a}{N}f\left ( a + k\left (\frac {b-a}{N}\right )\right )$ is a bit intimidating, but it is so useful that, we will see, there is a different way to notate this, as well as a surprisingly quick way to calculate this sum!