Find the mass of a 20-inch rod whose cross section is a 2 × 2 inch square, with density × inches from the left end given by $\rho ( x ) = 3.6 + 0.6 x - 0.03 x ^ { 2 }$ grams per cubic inch.
A) 440
B) 224
C) 446
D) 448

Question

Quizplus · Accepted Answer

The mass of the rod can be found by integrating the density function over the volume of the rod. The volume element $dV$ for a small segment $dx$ of the rod is $2 \times 2 \, dx = 4dx$ cubic inches. The mass element $dm$ is the density $\rho(x)$ times the volume element $dV$, so $dm = \rho(x) \cdot 4dx$. Integrating from $x = 0$ to $x = 20$ inches gives the total mass $M$:$$M = \int_{0}^{20} (3.6 + 0.6x - 0.03x^2) \cdot 4 \, dx$$$$= 4 \int_{0}^{20} (3.6 + 0.6x - 0.03x^2) \, dx$$$$= 4 \left[ 3.6x + 0.3x^2 - 0.01x^3 \right]_{0}^{20}$$$$= 4 \left[ 3.6(20) + 0.3(20)^2 - 0.01(20)^3 \right]$$$$= 4 \left[ 72 + 120 - 80 \right]$$$$= 4 \times 112 = 448$$Thus, the mass of the rod is 448 grams.

Find the Mass of a 20-Inch Rod Whose Cross Section ρ(x)=3.6+0.6x−0.03x2\rho ( x ) = 3.6 + 0.6 x - 0.03 x ^ { 2 }ρ(x)=3.6+0.6x−0.03x2

Find the Mass of a 20-Inch Rod Whose Cross Section $\rho ( x ) = 3.6 + 0.6 x - 0.03 x ^ { 2 }$