elvishwitchxyp

2022-01-03

Why is the integral of $d\sqrt{1+{x}^{2}}$ simply equal to $\sqrt{1+{x}^{2}}$ ?

Bubich13

Beginner2022-01-04Added 36 answers

This is a generalisation of the Riemann integral:

Before we had:

$\int}_{a}^{b}dx=x{\mid}_{a}^{b$

Now we have

$\int}_{a}^{b}d\left(g\left(x\right)\right)=g\left(x\right){\mid}_{a}^{b$

${\int}_{a}^{b}d\left(g\left(x\right)\right)$ is somewhat like ${\int}_{a}^{b}{g}^{\prime}\left(x\right)dx$ , much like in probability:

If we have a continuous random variable X in ($mathscr\left\{F\right\},\mathbb{P}$ ), then

$P(X\le x)={\int}_{-\mathrm{\infty}}^{x}d{F}_{X}\left(x\right)$

If X has a pdf, then we have

$P(X\le x)={\int}_{-\mathrm{\infty}}^{x}{F}_{x}^{\prime}\left(x\right)dx={\int}_{-\mathrm{\infty}}^{x}{f}_{X}\left(x\right)dx$

Not all continuous random variables have pdfs so we can use

Same with expected value: Let g be a Borel-measurable function. Then

$E\left[g\left(X\right)\right]={\int}_{\mathbb{R}}g\left(x\right){F}_{X}^{\prime}\left(x\right)dx={\int}_{\mathbb{R}}g\left(x\right){f}_{X}\left(x\right)dx$

If X has a pdf, then we have

$E\left[g\left(X\right)\right]={\int}_{\mathbb{R}}g\left(x\right){F}_{X}^{\prime}\left(x\right)dx={\int}_{\mathbb{R}}g\left(x\right){f}_{X}\left(x\right)dx$

Before we had:

Now we have

If we have a continuous random variable X in (

If X has a pdf, then we have

Not all continuous random variables have pdfs so we can use

Same with expected value: Let g be a Borel-measurable function. Then

If X has a pdf, then we have

lovagwb

Beginner2022-01-05Added 50 answers

When f' is continuous on $[a,b],$

${\int}_{a}^{b}df\left(x\right)={\int}_{a}^{b}{f}^{\prime}\left(x\right)dx=f\left(b\right)-f\left(a\right)$

See more information about the relationship between Riemann Stieltjes integrals and Riemann integrals here.

See more information about the relationship between Riemann Stieltjes integrals and Riemann integrals here.

karton

Expert2022-01-09Added 613 answers

Owing to the uncertain integrals property

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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