Question

Find the area of the region bounded by the graphs of y = x, y = −x + 4, and y = 0.

Answer 1

The line y = x and y = -x + 4 intersect when at the point (2, 2).

Expresing y = -x + 4 in terms of x, we have x = 4 - y.

Thus, the area of the region bounded by the graphs of y = x, y = −x + 4, and y = 0 is given by

$\int\limits^2_0 {(y-(4-y))} \, dy = \int\limits^2_0 {(y-4+y)} \, dy \\ \\ = \int\limits^2_0 {(2y-4)} \, dy= \left[y^2-4y\right]_0^2 =|(2)^2-4(2)| \\ \\ =|4-8|=|-4|=4$

Therefore, the area bounded by the lines is 4 square units.