Explanation

(c)
Given, $\int_{n}^{n + 1}{f(x)dx = n^{2}}$
On putting $n = - 2, - 1,0,1,2,3$. we get
$\int_{- 2}^{- 1}{f(x)dx = 4,}\ \ \ \int_{- 1}^{0}{f(x)dx = 1,}$
$\int_{0}^{1}{f(x)dx = 0,}\ \ \ \int_{1}^{2}{f(x)dx = 1,}$
$\int_{2}^{3}{f(x)dx = 4,}\ \ \ \int_{3}^{4}{f(x)dx = 9,}$
$\therefore\int_{- 2}^{4}{f(x)dx = 4 + 1 + 0 + 1 + 4 + 9 = 19}$