I'm an engineer teaching mathematics for about 5 years now. I have a very strong problem solving skills and exposure to many topics in mathematics. I'm currently working as a mathematics faculty in Allen Career Institute and teaching students for competitive level mathematics. I'm also working as a tutor on many online platforms like Photomath, tutoreye etc.
Find the escape velocity $v_{0}$ that is needed to propel a rocketof mass $m$ out of the gravitational field of a planet with mass$M$ and radius $R .$ Use Newton's Law of Gravitation (see Exer-cise 5.4 .33 ) and the fact that the initial kinetic energy of $\frac{1}{2} m v_{0}^{2}$supplies the needed work.
Show that if $a > - 1$ and $b > a+1,$ then the followingintegral is convergent.$$\int_{0}^{\infty} \frac{x^{a}}{1+x^{b}} d x$$
Electric charge is distributed over the rectangle 0$\leqslant x \leqslant 5$ 2$\leqslant y \leqslant 5$ so that the charge density at $(x, y)$ is $\sigma(x, y)=2 x+4 y$ (measured in coulombs per square meter). Find the total charge on the rectangle.
Find the mass and center of mass of the lamina that occupies the region $D$ and has the given density function $\rho .$$D$ is the triangular region with vertices $(0,0),(2,1),(0,3) ;$ $\rho(x, y)=x+y$
Find the mass and center of mass of the lamina that occupies the region $D$ and has the given density function $\rho .$$D$ is bounded by $y=1-x^{2}$ and $y=0 ; \rho(x, y)=k y$
Find the mass and center of mass of the lamina that occupies the region $D$ and has the given density function $\rho .$$D$ is bounded by $y=x+2$ and $y=x^{2} ; \rho(x, y)=k x^{2}$