How to find the area between two curves using integration?

Here is the complete solution. **Step 1: Find intersection points.** Set $x^2 = 2x - x^2$: \begin{align*} x^2 &= 2x - x^2 \\ 2x^2 - 2x &= 0 \\ 2x(x - 1) &= 0 \end{align*} So $x = 0$ and $x = 1$ are the intersection points. These become our limits of integration. **Step 2: Determine which curve is on top.** On the interval $[0, 1]$, pick a test point, say $x = 0.5$: - $y_1 = x^2 = 0.25$ - $y_2 = 2x - x^2 = 1 - 0.25 = 0.75$ Since $0.75 > 0.25$, $f(x) = 2x - x^2$ is the upper curve and $g(x) = x^2$ is the lower curve. **Step 3: Set up and evaluate the integral.** \begin{align*} A &= \int_0^1 [(2x - x^2) - x^2] \, dx \\ &= \int_0^1 (2x - 2x^2) \, dx \\ &= 2\int_0^1 (x - x^2) \, dx \\ &= 2\left[\frac{x^2}{2} - \frac{x^3}{3}\ ight]_0^1 \\ &= 2\left(\frac12 - \frac13\ ight) = 2\left(\frac{1}{6}\ ight) = \frac13 \end{align*} **General Strategy:** 1. Find all intersection points of $f(x)$ and $g(x)$. 2. Determine which function is larger on each subinterval (test a point). 3. Integrate $(\ ext{upper} - \ ext{lower})$ over each subinterval. 4. Add the absolute areas if curves cross within the interval. **For curves that cross at $x = c$:** $$A = \int_a^c [f(x) - g(x)] \, dx + \int_c^b [g(x) - f(x)] \, dx$$ This ensures we always subtract the smaller function from the larger one.