Animated Solution for Mathematics - Definite Integration: The area bounded by the curves $y=|x-1|+|x-2|$ and $y=3$ is equal to

The function $y=|x-1|+|x-2|$ represents the sum of distances from a point $x$ to the points $1$ and $2$ on the number line. The critical points, or "hinges," occur where the expressions inside the absolute value bars vanish, specifically at $x=1$ and $x=2$.

To understand the behavior of the function, we analyze it across three distinct intervals:

When $x<1$, both terms are negative. We flip the signs to obtain:

When $x\ge 2$, both terms are positive. We simplify to obtain:

In the interval $1\le x<2$, the first term is positive while the second is negative. The expression becomes: The $x$ terms cancel out, resulting in a perfectly horizontal line at $y=1$.

We now intersect this "boat-shaped" graph with the line $y=3$.

For the left branch:

For the right branch:

The vertices of the region bounded by the function and the line $y=3$ are $(0,3)$, $(1,1)$, $(2,1)$, and $(3,3)$.

The region formed is a trapezium. The parallel sides are the top segment (length $3-0=3$) and the bottom segment (length $2-1=1$). The height of the trapezium is the difference in $y$-values: $3-1=2$.

Using the area formula for a trapezium:

The final area is 4.