2016 Paper 2 Question 12 11th Oct 20201st Nov 2020nstmathsupervisor (a) Consider , then , so The polar angle is , thus the cylindrical polar coordinates are (b) Let Then Share this:TweetMore Email a link to a friend (Opens in new window) Email Like Loading... Related
![\begin{aligned} V &=\int dV \\ &= \int_{z=0}^{a} \int_{\theta=0}^{\alpha} \int_{r=0}^{r_z} r dr d\theta dz \\ &= \int_{\theta=0}^{\alpha}d\theta \int_{z=0}^{a} \left[ r^2/2 \right]_0^{r_z} dz \\ &= \frac{\alpha}{2} b^2 \int_{z=0}^{a} 1-\frac{z}{a} dz \\ &=\frac{1}{2}\alpha b^2 \left[ z - \frac{z^2}{2a} \right]_0^{a} \\ &= \frac{1}{4} \, \alpha \, a \, b^2 \end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D+V+%26%3D%5Cint+dV+%5C%5C+%26%3D+%5Cint_%7Bz%3D0%7D%5E%7Ba%7D+%5Cint_%7B%5Ctheta%3D0%7D%5E%7B%5Calpha%7D+%5Cint_%7Br%3D0%7D%5E%7Br_z%7D+r+dr+d%5Ctheta+dz+%5C%5C+%26%3D+%5Cint_%7B%5Ctheta%3D0%7D%5E%7B%5Calpha%7Dd%5Ctheta+%5Cint_%7Bz%3D0%7D%5E%7Ba%7D+%5Cleft%5B+r%5E2%2F2+%5Cright%5D_0%5E%7Br_z%7D+dz+%5C%5C+%26%3D+%5Cfrac%7B%5Calpha%7D%7B2%7D+b%5E2+%5Cint_%7Bz%3D0%7D%5E%7Ba%7D+1-%5Cfrac%7Bz%7D%7Ba%7D+dz+%5C%5C+%26%3D%5Cfrac%7B1%7D%7B2%7D%5Calpha+b%5E2+%5Cleft%5B+z+-+%5Cfrac%7Bz%5E2%7D%7B2a%7D+%5Cright%5D_0%5E%7Ba%7D+%5C%5C+%26%3D+%5Cfrac%7B1%7D%7B4%7D+%5C%2C+%5Calpha+%5C%2C++a+%5C%2C++b%5E2++%5Cend%7Baligned%7D&bg=ffffff&fg=333A42&s=0&c=20201002)
NST First Year Maths 