Answer to Question #339415 in Calculus for alli

Question

Find the moment about the x-axis of a wire of constant density that lies along The curve

𝑦 = √𝑥 from 𝑥 = 0 𝑡𝑜 𝑥 = 2

Answer 1

$M_x=ρ \int_a^b \frac 1 2 ([f(x)]^2-[g(x)]^2) dx$

where

$a=0, b=2$ $f(x)=\sqrt {x}, \space g(x)=0$ $ρ - constant \space density$

then

$M_x=ρ \int_0^2 \frac 1 2 x dx=\frac ρ 2 \int_0^2 x dx=\frac ρ 2 \frac {x^2} 2 \bigg|^2_0=\frac ρ 2 \frac 4 2=ρ$

Answer: the moment about the x-axis is equal to the constant density: $M_x=ρ$