For the region bounded by the line $y = x + 2$, the $x$-axis and the line $x = 2$, which of the following statements are true?

I. It is the triangle with vertices $(-2, 0)$, $(2, 0)$ and $(2, 4)$.

II. Its area is $\int_{-2}^{2} (x + 2)\,dx$.

III. With horizontal strips, its area is $\int_0^{4} (4 - y)\,dy$.

IV. With horizontal strips, its area is $\int_0^{4} (y - 2)\,dy$.

Explanation

1. 01

   The line meets the $x$-axis at $(-2, 0)$ and $x = 2$ at $(2, 4)$: a triangle of area $\frac{1}{2} \times 4 \times 4 = 8$, and $\int_{-2}^{2} (x + 2)\,dx = 8$. I and II hold.

2. 02

   A strip at height $y$ runs from $x = y - 2$ to $x = 2$, length $4 - y$: $\int_0^{4} (4 - y)\,dy = 8$, so III holds. IV measures the strip from the $y$-axis to the line and gives $\int_0^{4} (y - 2)\,dy = 0$: false.