Area Between Curves

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The area between curves is a major application of definite integrals in calculus. It lets us measure the size of a region enclosed by two graphs instead of just the area under one graph. This idea appears in physics, economics, and engineering whenever one quantity exceeds another over an interval. Learning this topic strengthens both graph interpretation and integral setup skills.

To find the area between two curves, subtract the lower function from the upper function and integrate over the interval where they bound the region. If the curves are given as functions of $x$ , the basic setup is an integral from the left intersection point to the right intersection point. If the top and bottom curves switch places, the interval must be split so each integral stays positive. In some problems it is easier to integrate with respect to $y$ , using right minus left instead of top minus bottom.

Key Facts

Area between curves with respect to $x$ : $A = \int_a^b [f(x) - g(x)]\,dx$ , where $f(x)$ is above $g(x)$ .
Intersection points are found by solving $f(x) = g(x)$ .
If curves switch order, split the area at the crossing point where the order switches.
Area between curves with respect to $y$ : $A = \int_c^d [x_{\text{right}}(y) - x_{\text{left}}(y)]\,dy$ .
Area is always nonnegative, so use upper minus lower or right minus left on each interval.
A definite integral can be negative, but geometric area cannot, so check the graph before setting up the integral.

Vocabulary

Definite integral: A definite integral gives the accumulated signed area of a function over a specified interval.
Intersection point: An intersection point is a point where two graphs have the same x-value and y-value.
Upper function: The upper function is the graph with the larger y-value on the interval being used.
Lower function: The lower function is the graph with the smaller y-value on the interval being used.
Bounded region: A bounded region is a closed area enclosed by curves or lines.

Common Mistakes to Avoid

Using the wrong order in the integrand, such as lower minus upper, because this can produce a negative result instead of the actual area. Always identify which graph is on top over the interval.
Forgetting to find the intersection points first, because the limits of integration must match the region being enclosed. Solve $f(x) = g(x)$ before writing the integral.
Using one integral when the curves cross in the middle, because the top function may change. Split the interval at the crossing point where the order switches.
Assuming area and signed integral are the same thing, because a definite integral can be negative while area cannot. Check the graph and use positive pieces for geometric area.

Practice Questions

1 Find the area between $y = x + 2$ and $y = x^2$ from their intersection points.
2 Find the area enclosed by $y = 4 - x^2$ and $y = x^2$ .
3 A student sets up the area between y = 2x and y = x^2 on [0, 3] as a single integral. Explain whether this setup is correct and describe any change that is needed.