Chapter 7.CR, Problem 84CR

Insect Cannibalism In certain species of flour beetles, the larvae cannibalize the unhatched eggs. In calculating the population cannibalism rate per egg, researchers need to evaluate the integral

${\displaystyle {\int }_0^{A}c\left(t\right)dt}$,

where, $A$ is the length of the larval stage and $c\left(t\right)$ is the cannibalism rate per egg per larva of age $t$. The minimum value of $A$ for the flour beetle Tribolium castaneum is $17.6$ days, which is the value we will use. The function $c\left(t\right)$ starts at day $0$ with a value $0$, increases linearly to the value $0.024$ at day $12$, and then stays constant. Source: Journal of Animal Ecology. Find the values of the integral using

a. formula from geometry;

b. the Fundamental Theorem of Calculus.