The area between two curves is found by integrating their vertical or horizontal separation over the interval where the region is bounded. A graph is valuable because it shows which curve is above the other and whether the region must be split.
1. Vertical-slice formula
If \(y=f(x)\) lies above \(y=g(x)\) for \(a\le x\le b\), then
\[A=\int_a^b\bigl(f(x)-g(x)\bigr)\,dx.\]
The integrand is “top minus bottom,” so it represents a nonnegative vertical height.
2. Worked example: line and parabola
Find the area enclosed by
\[y=2x\qquad\text{and}\qquad y=x^2.\]
First find intersections:
\[2x=x^2\Longrightarrow x(x-2)=0\Longrightarrow x=0,2.\]
On \([0,2]\), the line is above the parabola because, for example, at \(x=1\), \(2x=2>1=x^2\). Therefore
\[\begin{aligned}A&=\int_0^2(2x-x^2)\,dx\\&=\left[x^2-\frac{x^3}{3}\right]_0^2\\&=4-\frac83=\frac43.\end{aligned}\]
3. Verify the region visually
Open the Graph Shading tool, graph 2*x and x^2, and shade the region between them from 0 to 2. The shaded picture should match the integral bounds and the top-minus-bottom order.
4. When the curves switch order
If \(f-g\) changes sign inside the interval, one integral of \(f-g\) gives signed area and may cancel. Split at every intersection:
\[A=\int_a^c|f(x)-g(x)|\,dx+\int_c^b|f(x)-g(x)|\,dx.\]
In practice, determine the upper curve separately on each subinterval and remove the absolute-value notation by using the correct order.
5. Horizontal slices
Some regions are easier to describe with x as a function of y. If \(x=R(y)\) is the right boundary and \(x=L(y)\) is the left boundary for \(c\le y\le d\), then
\[A=\int_c^d\bigl(R(y)-L(y)\bigr)\,dy.\]
Horizontal slices are useful when vertical slices would require several separate integrals.
6. Improper or unbounded regions
A region near a vertical asymptote may require an improper integral. A graph that appears to enclose a thin tail does not guarantee finite area. Replace the asymptotic endpoint with a variable limit and test convergence before reporting a number.
7. Numerical versus exact answers
Use exact intersections whenever possible. If the equations meet at non-elementary values, numerical root finding and numerical integration may be appropriate, but state the approximation and use enough precision. A graph helps choose starting intervals for the numerical solver.
Common mistakes
- Using the viewing-window edges as integration bounds.
- Subtracting bottom minus top and accepting a negative area.
- Missing an interior intersection where the curves switch order.
- Using x-integration when y-integration would be much simpler.
- Assuming every visually bounded-looking region has finite area.
- Rounding intersection values too early.
For a region that will be rotated, continue with disk, washer, and shell methods. For graph construction, see how to shade between two curves.