Introduction to derivatives and integrals question

Question

I have recently started studying calculus and I constantly find my self baffled by its notations. Specifically can somebody explain the difference between:

$$\int_{a}^{b} x^2 dx$$ $$\int_{}x^2 $$ $$\int_{a}^{b} x^2$$

and $$\frac{d}{dt}x^2$$ $$\frac{dx}{dt}x^2$$

Answer 1

Disclaimer: These are my interpretations of the notation, but other people may have other ways of writing things.

The first step is noting that

$$\int_a^bf(x)\ dx=\text{Area}$$

while

$$\int f(x)\ dx=\text{anti-derivative}$$

These are two separate concepts, bound together by the fundamental theorem of calculus:

$$\int_a^bf(x)\ dx=F(b)-F(a),\text{ where }F(x)=\int f(x)\ dx$$

Anti-derivatives are defined as follows:

$$\frac d{dx}F(x)=f(x)\implies F(x)=\int f(x)\ dx$$

It is, lastly, in my greatest opinion that $\int f(x)$ is unclear notation. A lack of that final $dx$ will cause you great misfortune, for example, what does the following mean?

$$\iint\frac xy\sin(z)=?$$

Does it mean to integrate with respect to $x,y$ or $z$? And in which order? Clearly, this expression makes no sense, but,

$$\iint\frac xy\sin(z)\ dx\ dz$$

Makes perfect sense.

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Derivatives, on the other hand, have their own tricky notations:

The following means to take the derivative with respect to $t$ for everything after the $\frac d{dt}$:

$$\frac d{dt}x^2$$

But if there is something in the numerator, like

$$\frac{dx}{dt}x^2$$

It means to take the derivative of the numerator with respect to $t$, then multiply it by whatever comes after.

Answer 2

In all, I saw $5$ mathematical expressions:

1. The first one is a definite integral of $x^2$ from $a$ to $b$.
2. No meaning as it contains no differential. If it contained a differential, it would be an indefinite integral or an antiderivative.
3. No meaning due to above reason.
4. Derivative of $x^2$ with respect to $t$
5. Product of $x^2$ with the derivative of $x$ with respect to $t$

These are what they actually are.

Answer 3

The users above have sufficiently explained the difference between the integral notations, so I won't go over it again, but you asked for an explanation of the difference between $\frac {dx}{dt} x^2$ and $\frac d{dt} x^2$.

Well let's say that $x = f(t) = 2t^2$. Then the evaluation of the first expression would be to differentiate $x$ with respect to $t$ and multiply it by $x^2$: $\frac{dx}{dt} \times x^2 =4tx^2$.

The second one is a bit harder to evaluate. In this one, we are differentiating $x^2$ with respect to $t$. However, $x$ is a function of $t$ and does not contain any terms of $t$ in it, so how do we differentiate it? Well, in this case, we can rewrite $x^2$ using the function of $t$, but in other cases, we can't just do that. Instead, we use a method called implicit differentiation.