Astraight rod has one end at the origin and the other end at the point (l,0) and a linear density given by λ=ax2, where a is a known constant and x is the x coordinate. since this wire is not uniform, you will have to use integrtation to solve this part. use m=∫l0dm to find the total mass m. find xcm for this rod. express your answer in terms of one or both of a and l.

It is given that a straight rod has one end at the origin (that is (0,0)) and the other end at the point (L,0) and a linear density given  by, where a is a known constant and x is the x coordinate.

Therefore, the infinitesimal mass is given as:

Therefore, the total mass will be the integration of the above equation as:

Therefore,

Now, we can find the center of mass, of the rod as:

Now, we have

x_{cm}=\frac{1}{\frac{aL^3}{3}}\int_{0}^{L}ax^3dx=\frac{3}{aL^3}\times [\frac{ax^4}{4}]_{0}^{L}

Therefore, the center of mass, is at: