Question

The specific heat of a substance at temperature $t^{\circ }C$ is $s=a{t}^2+bt+c$. Find the amount of heat required to raise the temperature of ${}^{\prime }{m}^{\prime }$ grams of a substance from $0^{\circ }C$ to $t^{\circ }C$.

Answer 1

Heat,

$H=\int dH={\int }_0^{t}msdt={\int }_0^{t}m(a{t}^2+bt+c)dt=m[a{\int }_0^t{t}^{2}dt+b{\int }_0^{t}tdt+c{\int }_0^{t}dt]H=m[a.{\left(\frac{t^3}{3}\right)}_0^t+b{\left(\frac{t^2}{2}\right)}_0^t+{(c.t)}_0^t]H=m(\frac{a}{3}.t^3+\frac{b}{3}.t^2+c.t)$