The triple integral that represents the mass of a solid right circular cylinder $T$ of radius $4$ and height $3$ given that the density varies directly with half the distance from the axis of the cylinder is :
This is an easy question but I got confused a bit about"density varies directly with half the distance from the axis of the cylinder" this will be density. Is means $r^=\sqrt{x^2+y^2}\cdot 1/2.$ If so, this answer is $$\int_0^{2\pi}\int_0^{4}\int_0^{3} r^3 dz dr d\theta$$ Is that correct?
It seems to me that to say that “the density varies directly with half the distance from the axis of the cylinder” means that the function that should be $\frac12\sqrt{x^2+y^2}$ (as you wrote), and then the integral should be$$\int_0^{2\pi}\int_0^4\int_0^3\frac12r^2\,\mathrm dz\,\mathrm dr\,\mathrm d\theta.$$