Science:Math Exam Resources/Courses/MATH103/April 2009/Question 06 (b)/Solution 1

The total number of seeds is the sum of all $\ m^{2}$ around the tree times the passend concentration of seeds.

Because the area around the tree is a ring, the field of seeds looks like a the function rotated around the tree (y-axis).

To calculate this integral, we use the shell method $\displaystyle \int 2\pi xh(x)dx$

For integration we use integration by parts.

${\begin{aligned}\int _{0}^{10}2\pi \underbrace {r} _{=u}50\underbrace {\sin \left({\frac {\pi r}{10}}\right)dr} _{dv}&=100\pi \left[\left.-r\cos \left({\frac {\pi r}{10}}\right){\frac {10}{\pi }}\right|_{0}^{10}+{\frac {10}{\pi }}\int _{0}^{10}\cos \left({\frac {\pi r}{10}}\right)dr\right]\\&=100\left[-10\cdot 10\cos(\pi )+0+10\left.\sin \left({\frac {\pi r}{10}}\right){\frac {10}{\pi }}\right|_{0}^{10}\right]\\&=10000.\end{aligned}}$

There are 10000 seeds around the tree.