Shading between two curves requires three pieces of information: the two boundary graphs, the interval over which the region is considered, and which graph lies above or to the right.
Worked example
Shade the region bounded by y=2x+3 and y=x^2.
1. Find the intersections
x^2 = 2x + 3 x^2 - 2x - 3 = 0 (x - 3)(x + 1) = 0
The curves intersect at x=-1 and x=3.
2. Identify the upper and lower curves
Test x=0. The line gives 3 and the parabola gives 0. Therefore y=2x+3 is the upper boundary and y=x^2 is the lower boundary on [-1,3].
3. Create the graphs
Add both functions in the Graphing Calculator and use different colors. Confirm that the visible intersections match the algebra.
4. Add the shaded region
Open Graph Shading, choose the two functions, and set the x-bound from -1 to 3. Select a color and opacity that leave both boundary curves visible.
When the curves switch order
If two curves intersect inside the selected interval, the upper and lower functions may change. Split the region into subintervals so the chosen boundary order remains valid.
Regions bounded with respect to y
Some regions are easier to describe with left and right boundaries x=f(y). In that case, choose a y-bound and identify the left and right curves rather than upper and lower curves.
Relation to area
For continuous curves with upper function u(x) and lower function l(x), the area is found from the integral of u(x)-l(x) over the interval. The shaded picture helps verify that the integrand and bounds match the intended region.
Common mistakes
- Choosing arbitrary interval endpoints instead of intersections.
- Reversing upper and lower functions.
- Using an interval where the curves cross again.
- Hiding the boundary lines with fully opaque shading.
- Assuming the same x-bound works for a region described more naturally with y.
Shade the worked example online →
To combine several conditions, continue with graphing a system of inequalities.