1. **Stating the problem:** We want to create an integration system in Desmos that fits a polygon under any function, such as $f(x)=mx+b$ or $f(x)=x^2$, to approximate the area under the curve. 2. **Formula and concept:** The integral of a function $f(x)$ from $a$ to $b$ is given by $$\int_a^b f(x)\,dx,$$ which represents the exact area under the curve between $x=a$ and $x=b$. 3. **Approximating with polygons:** To approximate this area, we use polygons (usually rectangles or trapezoids) that fit under the curve. This is the basis of Riemann sums. 4. **Steps to create the polygon system:** - Divide the interval $[a,b]$ into $n$ subintervals of equal width $\Delta x = \frac{b-a}{n}$. - For each subinterval, calculate the function value at a chosen point (left endpoint, right endpoint, or midpoint). - Construct rectangles with height equal to $f(x_i)$ and width $\Delta x$. - The sum of the areas of these rectangles approximates the integral. 5. **In Desmos:** - Define $n$ as the number of rectangles. - Define $a$ and $b$ as the interval limits. - Define $\Delta x = \frac{b-a}{n}$. - Define the points $x_i = a + i \Delta x$ for $i=0,1,...,n$. - For each $i$, draw a polygon (rectangle) with vertices at $(x_i,0)$, $(x_i,f(x_i))$, $(x_{i+1},f(x_i))$, and $(x_{i+1},0)$. 6. **Example for $f(x)=mx+b$:** - The rectangles will have heights $f(x_i) = m x_i + b$. 7. **Example for $f(x)=x^2$:** - The rectangles will have heights $f(x_i) = (x_i)^2$. 8. **Summary:** This method works for any continuous function and the polygon will fit the space under the curve, approximating the integral. This approach can be implemented in Desmos using sliders for $a$, $b$, and $n$, and defining the function $f(x)$ accordingly.