Answer
The total basal metabolism is $~2040~kcal$
Work Step by Step
We can find the total basal metabolism:
$\int_{0}^{24}R(t)~dt = \int_{0}^{24}[85-0.18~cos(\frac{\pi~t}{12})]~dt$
Let $u = \frac{\pi~t}{12}$
$\frac{du}{dt} = \frac{\pi}{12}$
$dt = \frac{12~du}{\pi}$
When $t = 0$, then $u = 0$
When $t= 24$, then $u = 2\pi$
$\int_{0}^{2\pi}(85-0.18~cos~u)~(\frac{12~du}{\pi})$
$=\int_{0}^{2\pi}(\frac{1020}{\pi}-\frac{2.16}{\pi}~cos~u)~du$
$=(\frac{1020}{\pi}~u-\frac{2.16}{\pi}~sin~u)\vert_{0}^{2\pi}$
$=[\frac{1020}{\pi}~(2\pi)-\frac{2.16}{\pi}~sin~2\pi]-[\frac{1020}{\pi}~(0)-\frac{2.16}{\pi}~sin~0]$
$= (2040-0)-(0)$
$= 2040$
The total basal metabolism is $~2040~kcal$
