As stated in my previous entry, I like introducing Calculus from the Integral side. My students are already familiar with areas of circles and volumes of cylinders but curiosity still persists with respect to why the volume of a cone is what it is. The first two days are thus surrendered to discussion revolving around this topic.

Sigma notation is first discussed and the formula for sum of triangular numbers is quickly derived. I also mention that the formula for sum of squares is required in our search for the cone’s curious volume. They want to see how it is derived; enter telescoping sums. Through these procedures, students are introduced to new notation and gain insights into how concepts they’ve already learned are exploited in the search for bigger things.

Once the sum formulas are in place, the interval [0,h] along the x-axis is divided into sub-intervals of uniform length. This length determines the “height” dimension of each cylindrical slab; the radius is based on the upper bound of each partition. It is clear to all that this radius is variable and is governed by f(x). After some scribbles and factoring, the need for sum of squares formula surfaces and soon thereafter, so does the cone’s volume.

After a few examples of setting up the interval along the x-axis, I turn things on their side and have students find the same volume using f(y) as radius. Directly below, you will find such an example. The link above the image will open a new tab, providing students with a little interactive exploration. Beneath my slab method, I’ve included the same for Cylindrical Shells.

I consider integration a tool for much better things that will be pursued throughout the semester. My students will be fluent with integration and differentiation by the end of February; then the real fun begins.

Cylindrical Slabs

Cylindrical Shells

Also check out volume of sphere.

Thanks for reading.

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