A "signed definite integral" for computing work and other "net change" calculations. The value of an expression such as $\int_0^1 x^2\,dx$ comes out the same under all these interpretations, …

Aug 21, 2023 · 不定积分作为工具刚开始使用时，没有人想到它其实和面积能扯上关系，只是单纯地作为微分的逆映射。直到后面的定积分。 此时单纯善良的人们还不知道曲边梯形的面积居 …

The symbol used for integration, $\int$, is in fact just a stylized "S" for "sum"; The classical definition of the definite integral is $\int_a^b f (x) dx = \lim_ {\Delta x \to 0} \sum_ {x=a}^ {b} f …

Jun 21, 2014 · 概念的话dx就是把定义域的x范围无限分（微分）其中的一份如x1 到x2 这一小段就是dx。 同理，dy就是值域的无限分为f (x2)-f (x1)。 dy/dx 是f (x)一个微分成dx dy围成的小三 …

May 2, 2015 · Okay this may sound stupid but I need a little help... What do $\Large \frac {d} {dx}$ and $\Large \frac {dy} {dx}$ mean? I need a thorough explanation. Thanks.

Dec 27, 2013 · Well, $\delta x$ means different things depending on the context. For example, it has a particular meaning in variational calculus, and a completely different one in functional …

Jan 12, 2016 · These identities for $\int_0^1 x^ {-x}\ dx$ and $\int_0^1 x^x\ dx$ are sometimes called the "sophomore's dream". Look that up on Wikipedia.

I am working on trying to solve this problem: Prove: $\\int \\sin^n{x} \\ dx = -\\frac{1}{n} \\cos{x} \\cdot \\sin^{n - 1}{x} + \\frac{n - 1}{n} \\int \\sin^{n - 2}{x ...

I just finished taking my first year of calculus in college and I passed with an A. I don't think, however, that I ever really understood the entire $\\frac{dy}{dx}$ notation (so I just focused on ...

Here, $ (dx)^2$ means $dx \wedge dx$, and the fact that it vanishes comes from the fact that the exterior algebra is anti-commutative. In other words, formally we have $d^2x=0$ and $ …