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Section 1.9 Shredding to find areas

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Worksheet
A4US

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One of the big questions we can address using calculus is the calculation of areas, particularly the area of weirdly shaped objects. In this exercise we will explore the idea of calculating an area using one of the principles we have for the course, "When you don't know the answer, make as good a guess as you can. Then, try to figure out how to make your guess better."

We'll be trying to find the area of the shaded area shown to the right. Let me assure you that there is no simple equation that can describe the curve nor is there really a way to find the exact answer - our goal is to approximate the area as best we can.

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1.

With your group, brainstorm as many ideas as possible that answer the question, "How might we measure the area inside this curve?" You may have encountered answers to this question in your previous studies - for example the technique of using Riemann sums. Certainly put those on your list, but also try to be as creative as possible!

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Looking at the list of strategies, each group should pick one or two and discuss how you might iterate the technique to get a better estimate.

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The instructor will distribute an envelope and a ruler to each group.

Inside the envelope, you'll find strips of paper that are the result of shredding an image like the one we've been working with. Using only the ruler, create two estimates of the area - an over- and underestimate using the framework of Riemann sums. Then, write down the Riemann sums that represent your estimates. How could you refine them?

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4.

Let's look at the definition of a Riemann sum for a function f(x) on the interval [a,b] with n rectangles,

n∑i=1f(x∗i)b−an.
  • Why do you think Riemann chose this particular method for estimating areas? Why use rectangles? Why make all of them the same width? Why use right and left hand endpoints?
  • Does Riemann's method always work? In other words, if we keep increasing the number of rectangles, will we get better and better estimates of the area? Can you think of a function where the estimation fails?