There aren’t that many problems that I wrote in my first year and that have survived untouched since that time. But this collection of area problems regularly puts a smile on my face when I get to it. These problems are simple, but they do a surprisingly good job of getting students to think about the connections between antiderivatives and signed area.
The first problem has students connect antiderivatives to geometric determinations of area and introduces the idea of “positive” and “negative” areas canceling each other.
The second connects the idea of integration to a familiar area formula, specifically making the connection between a circle’s area and its perimeter. You can visualize this by drawing a Riemann sum on top of a triangle and then “unwrapping it” to make a circle (as in this post on the Better Explained blog). I’ve built a physical model of this using plastic tubes, which can be bent to fill in either a semicircle or a triangle:
The third problem has students revisit the idea of canceling positive and negative areas, in a context where the geometric approach is less intuitive than in question 1. You can discuss this problem by implicitly invoking the IVT, even if you don’t name it: If a=1, we only have positive area. But if a gets big enough, the negative area clearly overwhelms the positive area. So there must be some value of a between 1 and “large” at which the positive and negative areas cancel. Their job is to find that value.
The fourth again asks them to think geometrically, but in an approximate way: does it feel plausible that the area formed by a parabola occupies 2/3 of the triangle containing it? At a quick glance, it does.
The fifth asks students to find area using an antiderivative, then think about it in terms of the reality of a graph. In this case, the area interpretation draws on the fact that the graph of \(\scriptsize y=x^n\) always goes through both (0,0) and (1,1) and that increasing n makes the graph closer to 0 between these two points, and farther from 0 after x=1.
The sixth (and last) problem has students evaluate an integral using a mix of area and antiderivative. Specifically, I give this problem before teaching students about the derivative and antiderivatives of trig functions, so they must handle the \(\scriptsize y=x^2\) part using an antiderivative, but use signed area to argue that the sine and cosine parts of all these integrals are equal to zero.
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