I'm interested in opinions based on experience about how to introduce volume for beginner calculus students. Below I present some observations and specific questions.
In Stewart's book, the volume of a solid $S$ that lies between $x=a$ and $x=b$ is defined by $$V=\int_a^b A(x)\ dx,$$ where $A(x)$ is the area of the vertical cross-sectional area of $S$ through $x$. This definition comes from vertical slicing.
In the examples of the said book, the following formula that comes from horizontal slicing for the volume of a solid $S$ that lies between $y=c$ and $y=d$ is also used: $$V=\int_c^d B(y)\ dy,$$ where $B(y)$ is the area of the horizontal cross-sectional area of $S$ through $y$.
It seems the said book gives no explanation of the fact that both formulas give the same result when applied to the same solid.
Do you introduce volume for beginners calculus students as defined in Stewart's book?
- If so, how do you deal with the fact that both formulas give the same number? Do you ignore this fact or assume that it is obvious? Why?
- If not, what is your approach? Do you give a definition or just integral formulas? Why?