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

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This question is remarkably close to one of the first questions we posed this semester: how do you determine the exact value of , 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?

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; 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!