Articles of calculus

Calculate center of mass multiple integrals

Can you help me with this problem? Find the center of mass of a lamina whose region R is given by the inequality: and the density in the point (x,y) is : The region r is this one: Is this the proper way to set up the integral for m: $$\int_{-1}^{1}\int_{-x-1}^{x+1} \ e^{x+y} \ dy […]

Derivative of integral with x as the lower limit

Question: Let $$F(x) =\int_{x^3}^{5}(cos^2t-te^t)dt $$ Find $F'(x)$ We were not explicitly taught about this during the semester but from what I can gather from online readings is that $$F'(x)=F'(5)-F'(x^3)$$ Therefore, $$F'(x)=(cos^2(5)-5e^5)-(cos^2(x^3)-x^3e^{x^3})$$ Is this correct? Thank you for any help!

Show that $\left | \sin a – \sin b \right | \leq \left | a-b \right |$

Show that $$\left | \sin a – \sin b \right | \leq \left | a-b \right |$$ I saw a proof using The Mean Value Theorem, but I could not grasp it well. Are there any other proofs or can someone clarify it by using The Mean Value Theorem?

Given point and tangent line

The function is $f(x)=1/x^2$. I have to find equation of the tangent line that also goes through the point $(0,12)$. I know how to construct this problem in desmos: So by reading the graph I know solutions are: $y=-16x+12$ $y=16x+12$ But how do I find this equations only by calculation? How do I figure […]

Show that: $\lim\limits_{n\to\infty} \sqrtn = 1$

First of all, I already know the common proof for this limit. My question concerns a specific proof that I could not deal with. It starts with defining a sequence $\{x_n\}_{n=1}^{\infty}$ with the general term $x_n=\sqrt[n]{n}-1$ and then shows that this sequence converges to $0$. We have $n=(1+x_n)^n\geq \frac{n(n-1)}{2}x_n^2$ Up to this point everything is […]

Mollifiers: Asymptotic Convergence vs. Mean Convergence

Problem Does asymptotic convergence imply mean convergence: $$\varphi\in\mathcal{L}_\text{loc}(\mathbb{R}_+):\quad\varphi(T)\stackrel{T\to\infty}{\to}\varphi_\infty\implies\frac{1}{T}\int_0^T\varphi(s)\mathrm{d}s\stackrel{T\to\infty}{\to}\varphi_\infty$$ Remark Three important classes fall under local integrability: $\mathcal{L}(\mathbb{R}_+),\mathcal{C}(\mathbb{R}_+),\mathcal{B}(\mathbb{R}_+)\subseteq\mathcal{L}_\text{loc}(\mathbb{R}_+)$

Maxima problem?

this question is off a textbook, and I’ve been having a lot of trouble with it: “A window consists of a rectangle surmounted by a semi-circle having its diameter the width of the rectangle. If the perimeter of the window is $t$ meters, find the greatest possible area of the window.” The final answer is: […]

The solution of $\min_x \max_{1\leq r \leq N} \left|\frac{\sin rMx}{\sin rx}\right|$

I am looking for the solution of $$\min_x \max_{1\leq r \leq N} \left|\frac{\sin rMx}{ M \sin rx}\right|$$ where $M < N$ are integers and $x \in \mathbb{R}^+$. For $M = 4, N = 6$, $f_{r,M}(x) =\left|\frac{\sin rMx}{ M \sin rx}\right|$ is plotted for different values of $r$ in the figure below. The maximum is plotted […]

Power series expansion

I recently had a problem. I know how to evaluate power series but I cannot seem to find an expansion for $\sqrt{x+1}$. I’ve tried differentiating it, in order to bring it in reciprocal form but that didn’t help. Due to the presence of square root, I cannot change it in the form of $1/(x+1)$. Kindly […]

Evaluate $\int \dfrac{1}{\sqrt{1-x}}\,dx$

Find $$\int \dfrac{1}{\sqrt{1-x}}\,dx$$ I did this and got $\dfrac23(1-x)^{\frac32} + c$ But a online calculator is telling me it should be $-2(1-x)^{\frac12}$ What one is on the money and if not me why?