Articles of integration

Simple algebra in a differential equation.

I have the differential equation: $$\frac{dy}{dx}=\sin (x-y).$$ Substituting $v=x-y$ and $dy=dx-dv$, I got down to the equation:$$\frac{dv}{1-\sin(v)}=dx.$$ Multiplying the LHS by $\dfrac{1+\sin (v)}{1+\sin(v)}$, I got:$$(\sec^2(v)+\tan (v)\sec (v))dv=dx.$$ This is an easy integral, and I got that $x=\tan(v)+\sec(v)$, with some constant of integration. Now, doing this in Maple gives the result: $$\frac{2}{1-\tan(\frac{v}{2})}=x,$$ where there should be […]

Find the volume of rotation about the y-axis for the region bounded by $y=5x-x^2$, and $x^2-5x+8$

Find the volume of rotation about the y-axis for the region bounded by $y=5x-x^2$, and $x^2-5x+8$ Here is an image: Normally I can do this question, but this one is tricky because since we are rotating about the y-axis, and we are quadratic, when I solve for $x$ I get two answers, one positive and […]

Checking proof that a given process is a martingale

I am interested in justify the well known result about the process $M^\lambda _t =\exp\left(\lambda B_t – \frac{\lambda^2}{2} t\right)$ being $\mathcal F_t$-martingale in the filtered probability space $(\Omega,\mathcal F, (\mathcal F_t), \mathbb P)$ where $(B_t)$ is a standard Brownian motion and $(\mathcal F_t)$ is it’s canonic filtration. I starte by noting that since $M^\lambda > […]

Necessary and sufficient condition for Integrability along bounded variation

Related: When is it that $\int f d(g+h) \neq \int f dg + \int f dh$? In this context, I write “integration” to mean the Riemann-Stieltjes integeation Let $g:[a,b]\rightarrow \mathbb{R}$ be of bounded variation. Let’s define $\alpha(x)=V_a^x(g)$ and $g_1(x)= V_a^x (g) – g(x)$ and $g_2(x)= V_a^x(g) + g(x)$. ($V_a^x$ means the total variation) Then, these […]

Techniques for evaluating probability integral

Consider the integral of a normal distribution: $$\int_a^b f(x)\,\mathrm d x=c $$ and a second integral for the expected value: $$ \int_a^b x\cdot f(x)\,\mathrm dx $$ Since you know the first integral is equal to $c$, what is a good way to evaluate the second integral to find the expected value? Integration by parts doesn’t […]

If $f$ is Riemann-Stieltjes Integrable, then does there exist a partition of which each lengths of subinterval are the same?

Let $\alpha$ be a monotonically increasing function. Say, $f\in\mathscr{R}(\alpha)$. Then does there exist a partition $P=\{x_0,…,x_n\}$ such that $$x_i=a+ \frac{b-a}{n}i,$$ $i\in\{0,\ldots,n\}$ and $$U(P,f,\alpha)-L(P,f,\alpha)<\epsilon$$ for each $\epsilon>0$?

Derivative under a double integral

How does one find ${\partial y\over \partial t}$ and ${\partial^2 y\over \partial t^2}$ of a double integral $$y(x,t)=\int\limits_0^t \int\limits_{x-t+\xi}^{x+t-\xi} F(\eta)\,\,\,d\eta \,\,\,d\xi$$?

Length of a curve y = 1 – √x

I want to know the length of the curve described by $$f(x) = 1 – \sqrt{x},\quad x \in [0,1].$$ When I build the derivative and plug it in the length formula: $$\int_0^1 \sqrt{1 +\frac{1}{4x}} dx = x+\frac{\log(1)}{4} – x+\frac{\log(0)}{4}$$ I get a problem because of $\log(0)$. I have no idea what to do now. Thanks […]

partial integration

I have a short question about partial integration. If I want to determine an integral of the form $\int f’gdx$, the formula for partial integration is: $$\int f’gdx=[fg]-\int fg’dx$$ https://en.wikipedia.org/wiki/Integration_by_parts . Sometimes it is useful to apply the integration rule twice, for example if $g=x^2$ and then you have to apply partial integration on $\int […]

Prove surface area of a sphere using solid of revolution surface area formula.

I have to prove the surface area of a sphere with $r=1$ using the solids of revolution through revolution abouth both the $x$ and the $y$ axis. The formulas are easy. From top to bottom, surface area of revolution about $x$ axis, and $y$ axis formulas: $$S_x=\int_a^b2\pi y\,\sqrt{1+\Big(\frac{dy}{dx}\Big)^2}\,dx$$ $$S_y=\int_a^b2\pi x\,\sqrt{1+\Big(\frac{dx}{dy}\Big)^2}\,dy$$ Where in the first formula, […]