Area

Table of Contents

Understanding Area via Integrals

In this chapter we focus on one of the most important applications of definite integrals: computing area. We will assume you already know the basic idea of definite integrals and the Fundamental Theorem of Calculus from earlier chapters in Integral Calculus.

Our goal is to connect the geometric idea of “area” with the analytic tool of integration, and to learn standard setups you will use again and again.

1. Area Under a Curve and the $x$-Axis

The most basic situation: the area between the graph of a function and the $x$-axis on some interval.

Assume:

$f$ is continuous on $[a,b]$,
$f(x) \ge 0$ for all $x$ in $[a,b]$.

Then the area of the region bounded by:

the graph $y = f(x)$,
the $x$-axis,
the vertical lines $x = a$ and $x = b$,

is given by
$$
\text{Area} = \int_a^b f(x)\,dx.
$$

Here the integral directly equals the geometric area, because $f(x)$ is nonnegative.

Signed area vs. geometric area

In general, a definite integral gives a signed area:

If $f(x) \ge 0$ on $[a,b]$, then $\int_a^b f(x)\,dx$ is the (positive) area between the curve and the $x$-axis.
If $f(x) \le 0$ on $[a,b]$, then $\int_a^b f(x)\,dx$ is negative, with magnitude equal to the area between the curve and the $x$-axis.
If $f$ changes sign, the integral over $[a,b]$ is “area above the axis minus area below the axis.”

When you are asked for “the area of the region,” you must make it nonnegative. That leads to:

2. Area Between a Curve and the $x$-Axis When the Curve Crosses the Axis

Suppose $f$ is continuous on $[a,b]$ but sometimes positive and sometimes negative. To find the geometric area between $y = f(x)$ and the $x$-axis, you must:

Find all points in $[a,b]$ where $f(x) = 0$ (the $x$-intercepts).
Use these $x$-values to split $[a,b]$ into subintervals on which $f$ does not change sign.
On each subinterval where $f(x)\ge 0$, the contribution to area is $\int f(x)\,dx$.
On each subinterval where $f(x)\le 0$, the area contribution is $-\int f(x)\,dx$ (the negative of the integral).
Add all these positive contributions.

Symbolically, you are computing
$$
\text{Area} = \int_{a}^{b} |f(x)|\,dx,
$$
but in practice you usually split the interval where $f(x)$ changes sign and use separate integrals.

3. Area Between Two Curves (Horizontal Slices)

Next, consider the area of a region bounded between two graphs:

$y = f(x)$ (the “top” curve),
$y = g(x)$ (the “bottom” curve),
and the vertical lines $x = a$ and $x = b$.

Assume:

$f$ and $g$ are continuous on $[a,b]$,
and $f(x) \ge g(x)$ for all $x$ in $[a,b]$.

Then the area of the region between them is
$$
\text{Area} = \int_a^b \bigl(f(x) - g(x)\bigr)\,dx.
$$

This comes from thinking of the region as stacked vertical strips (“slices”) of width $dx$ and height $f(x)-g(x)$.

If the curves cross

If $f$ is not always above $g$ on the whole interval, then you must:

Find all intersection points where $f(x) = g(x)$ in $[a,b]$.
Use these points to split $[a,b]$ into subintervals.
On each subinterval, determine which function is on top.
Integrate “top minus bottom” on each subinterval, then add all resulting positive areas.

4. Area with Respect to $y$ (Vertical vs. Horizontal Slicing)

Sometimes regions are easier to describe using horizontal slices instead of vertical ones. In that case, you integrate with respect to $y$.

Assume a region is bounded by:

a right curve $x = X_{\text{right}}(y)$,
a left curve $x = X_{\text{left}}(y)$,
and horizontal lines $y = c$ and $y = d$.

Assume $X_{\text{right}}(y) \ge X_{\text{left}}(y)$ for all $y$ in $[c,d]$. Then the area is
$$
\text{Area} = \int_c^d \bigl(X_{\text{right}}(y) - X_{\text{left}}(y)\bigr)\,dy.
$$

Here each slice is a horizontal strip of height $dy$ and length $X_{\text{right}}(y) - X_{\text{left}}(y)$.

Choosing vertical vs. horizontal slices

You choose the direction of slices to make:

the bounds simpler,
and the integrand easier to express.

In many problems, you will try both points of view mentally and pick the simpler integral.

5. Typical Setups for Area Problems

When solving area problems, there is a general procedure you can follow:

Draw a clear diagram.
Sketch the curves, label axes and intersection points as best as you can.
Identify what region is being described.
Locate all bounding curves and lines (axes or constants like $x = 1$, $y = 3$, etc.).
Decide on vertical or horizontal slicing.

Vertical slices: area as $\int (\text{top} - \text{bottom})\,dx$.
Horizontal slices: area as $\int (\text{right} - \text{left})\,dy$.

Find intersection points and bounds.

For vertical slices: solve for $x$-values where curves meet; those become the $x$-limits.
For horizontal slices: solve for $y$-values where curves meet; those become the $y$-limits.

Write the integrand as “outer minus inner” or “top minus bottom.”
Ensure that the quantity inside the integral is nonnegative on the interval.
Evaluate the integral using antiderivatives.

This procedure is the same idea whether you are finding area under a single curve or area between two curves.

6. Area and Symmetry

Symmetry can simplify many area calculations:

Even functions: If $f$ is even, i.e. $f(-x) = f(x)$, then
$$
\int_{-a}^a f(x)\,dx = 2\int_0^a f(x)\,dx.
$$
So the area from $-a$ to $a$ is twice the area from
Odd functions: If $f$ is odd, i.e. $f(-x) = -f(x)$, then
$$
\int_{-a}^a f(x)\,dx = 0,
$$
which tells you that the positive area on one side of the origin equals the negative area on the other side. For geometric area, you might then double the positive part:
$$
\text{Area} = 2\int_0^a |f(x)|\,dx
$$
when $f(x)$ has a definite sign on $[0,a]$.

Symmetry also appears when regions are symmetric about an axis or a line; in many cases, you can find the area of one symmetric part and multiply by 2 (or 4, etc.).

7. Piecewise and Composite Regions

Some regions are built from multiple simpler pieces. In such cases, you can:

Split the region into subregions for which you can write simple integrals.
Compute each area separately.
Add or subtract these areas as needed.

For example, if a region naturally breaks at some $x = c$ or $y = k$ where the bounding curves change, you can write
$$
\text{Total Area} = \int_a^c (\text{top}_1 - \text{bottom}_1)\,dx
+ \int_c^b (\text{top}_2 - \text{bottom}_2)\,dx
$$
or the analogous form with $dy$.

The key idea is that area adds: if $R$ is the union of non-overlapping regions $R_1$ and $R_2$, then
$$
\text{Area}(R) = \text{Area}(R_1) + \text{Area}(R_2).
$$

8. Area in Terms of Riemann Sums (Conceptual Link)

While the Fundamental Theorem of Calculus is used to evaluate area integrals, the connection to area originally comes from Riemann sums:

Divide the interval $[a,b]$ into $n$ small subintervals.
On each subinterval, approximate the area by a rectangle whose height is given by the function value at some point in the subinterval.
Add the areas of all these rectangles.

The definite integral
$$
\int_a^b f(x)\,dx
$$
is defined as the limit of these sums as the rectangles become thinner and more numerous. When $f(x)\ge 0$, this limiting process produces the exact geometric area.

This idea underlies all area formulas you write using integrals; even when the formulas look different (top minus bottom, right minus left), they all come from adding up many thin strips.

9. Common Types of Area Questions

Here are the main categories of area problems you will encounter:

Area under/above a single curve on an interval, possibly with the curve crossing the axis.
Area between two curves given as $y$-functions of $x$.
Area between two curves where it is simpler to rewrite them as $x$-functions of $y$ and integrate with respect to $y$.
Area of regions bounded by curves and lines, such as between a curve and the line $x = a$ or $y = b$.
Area involving symmetry, where you exploit symmetry to reduce computation.

In each case, the essential decisions are:

which variable to integrate with respect to,
what the slice looks like (vertical or horizontal),
what the bounds are,
and which function is “outer,” “inner,” “top,” or “bottom” on those bounds.

10. Connection to Later Topics

The techniques for setting up area integrals reappear in more advanced applications:

In volume problems (by slices, disks, washers, or shells), you use very similar reasoning, just in three dimensions.
In probability and statistics, integrals of density functions over intervals give probabilities; the computation looks like an area problem.
In physics, integrals that compute work, mass, or charge often use the same idea of slicing a region and summing contributions.