find the area of the region in two ways. a. using integration with respect to x. b. using geometry.

Explanation:

Step1: Find area using integration

We need to find the intersection points of the lines. The lines are $y = 2 - x$ and $y=x$. Setting $2 - x=x$, we get $2x = 2$, so $x = 1$. The region is bounded by $y = 2 - x$, $y=x$ and $x = 0$. The area $A$ using integration with respect to $x$ is given by $A=\int_{0}^{1}((2 - x)-x)dx=\int_{0}^{1}(2 - 2x)dx$. [ \begin{align*} \int_{0}^{1}(2 - 2x)dx&=\left[2x-x^{2}\right]_{0}^{1}\ &=(2\times1 - 1^{2})-(2\times0-0^{2})\ &=2 - 1\ &=1 \end{align*} ]

Step2: Find area using geometry

The region is a triangle. The base and height of the triangle are both 1. The area of a triangle is $A=\frac{1}{2}\times base\times height$. Here, base = 1 and height = 1, so $A=\frac{1}{2}\times1\times1 = 1$.

Answer:

1