<s>[INST] question about calculus I want to find the difference in volume between two balls, one with radius of x while the other with radius of $x+dx$. I find two answers, $\frac{4}{3}\pi(x+dx)^3-\frac{4}{3}\pi x^3$ and $4πx^2 dx$. Please tell me what I did wrong. Thanks.
Pardon my bad English.
 [/INST] Your answers are the same if we treat higher powers of $dx$ as being equal to $0$.
$$\begin{align}
\frac{4}{3}\pi(x+dx)^3-\frac{4}{3}\pi x^3 
&= \frac{4}{3}\pi(x^3+3x^2dx + 3x dx^2 + dx^3)-\frac{4}{3}\pi x^3 \\
&= \frac{4}{3}\pi(x^3+3x^2dx)-\frac{4}{3}\pi x^3 \\
&=\frac{4}{3}\pi x^3+4 \pi x^2dx-\frac{4}{3}\pi x^3 \\
&= 4 \pi x^2dx \\
\end{align}$$
Essentially, you're using the dual numbers approach to nonstandard analysis.
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