Q. 16 - Calculus

Question

Find the mass of a 30-centimeter rod with square cross sections of side length 2 centimeters, given that the density of the rod x centimeters from the left end is ρ(x) =$10·5+0·01527x^2$ grams per cubic centimeter.

Step-by-Step Solution

Answer

The mass of rod is $1809.72$ grams

Step 1: Given information

Density of material of rod is $\rho \left(x\right)=10.5+0.01527x^2$

length of rod is 30 centimeters

square cross section of side length 2 centimeters

Step 2: Calculation

Calculate the mass of rod 

$$\begin{gathered} dm=\rho \left(x\right)dv \\ \rho \left(x\right)=10·5+0.01527x^2 \\ dm=\rho \left(x\right)\left(2\times 2\right)dx \\ =4\left(10.5+0·01527x^2\right)dx \\ M={\int }_0^{30}dm \\ ={\int }_0^{30}4\left(10.5+0.01527x^2\right)dx \\ evaluate the integral \\ M=4{\left[10.5x+\frac{0.01527x^3}{3}\right]}_0^{30} \\ =1809.72 \end{gathered}$$

The mass of rod is $1809.72$grams