## Mathematics 12: Intermediate Calculus

Benjamin Hutz, Visiting Assistant Professor of Mathematics

In Math 12, the second semester of the introductory calculus sequence, students apply methods of integration to compute the volume of certain types of objects. In search of interesting objects to study, Math 12 set out to the Mead Art Museum at Amherst College. One of the requirements of the theory is that the object can be described by a curve rotated around a fixed line. The idea is that volume is computer by "adding up" the cross-sectional areas and, in the case of rotation, each cross-section is a disk. The area of a disk of radius** R** is easily computed as

**π**. The volume is then computed by "adding up" the disks through integration by treating the radius as a function of displacement. In other words:

*R*^{2}

** Area =_{ }π(R(x))^{2}dx**.

For example, if we rotate the curve ** y = x^{2}** defined from

**to**

*x =*0**about the**

*x =*1**we produce a bowl shaped object which is divided up into disks, see the figures below. We compute the area as:**

*y*-axis

_{}π(*x*^{2})^{2}*dx*= π_{}*x*^{4}*dx*=^{π}/_{5}### The Projects

By clicking the images below, you will find:

The work of art analyzedA Mathematica image of the curve that was rotated (i.e. the profile of the object).The total volume of the object