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: $M_x=ρ$