Areas between curves

Section 5.6 Areas between curves

We introduced Riemann sums in Section 2.1 and Section 5.2 as approximations of the area under the graph of a positive function. Then, we argued in Section 5.3 that by taking the limit where the number of rectangles becomes infinite, we obtain a precise calculation of the area. This is the fundamental idea behind Riemann sums and definite integrals.

🔗

In this section we study in more detail the intimate connection between definite integrals and areas between curves. In particular, we drop the assumption that the curve is given by the graph of a positive function. This is our first application of integration, in this case to geometry. We will see many more applications of integration in MATH 146.

🔗

Objectives

You should be able to:

Describe why the area bounded by the graphs of two functions can be written as a definite integral.
Write down and evaluate the definite integral representing the area of a given planar region.
Differentiate between geometrical situations where integration in $y$ is more appropriate than integration in $x$ to calculate the area of a given planar region.
Determine when it is necessary to split a given planar region into sub-regions and use more than one integral to evaluate the area.

🔗

Subsection 5.6.1 Instructional video

🔗

Subsection 5.6.2 Key concepts

🔗

Concept 5.6.1. Definite integrals and areas between curves.

The area $A$ of the region $S$ between the curves $y=f(x)\text{,}$ $y=g(x)\text{,}$ and the vertical lines $x=a$ and $x=b\text{,}$ with $a \leq b\text{,}$ is given by the definite integral

A = \int_{a}^{b} | f (x) - g (x) | d x .

$\begin{equation*} A = \int_a^b \left| f(x) - g(x) \right|\ dx. \end{equation*}$

This expression can be obtained by slicing the region into $n$ rectangles of equal width $\Delta x\text{.}$ The Riemann sum then gives an approximation of the area $S\text{,}$ and the limit $n \to \infty$ calculates $A\text{.}$ To recover the integral expression directly, draw a typical rectangle, with width $dx$ and height $|f(x) - g(x)|\text{:}$ the area is given by integrating over typical rectangles from $x=a$ to $x=b\text{.}$

If $f(x) \geq g(x)$ over the interval $[a,b]\text{,}$ the integral simplifies to

A = \int_{a}^{b} (f (x) - g (x)) d x .

$\begin{equation*} A = \int_a^b (f(x) - g(x) ) dx. \end{equation*}$

In the general case, where $f(x) \geq g(x)$ for some values of $x$ but $g(x) \geq f(x)$ for other values of $x\text{,}$ to evaluate the integral above we split the region $S$ into smaller subregions $S_1, S_2, \ldots, S_n$ where either $f(x) \geq g(x)$ or $g(x) \geq f(x)$ over a given subregion. Then we can use the fact that

| f (x) - g (x) | = {\begin{cases} f (x) - g (x) & if f (x) \geq g (x), \\ g (x) - f (x) & if g (x) \geq f (x), \end{cases}

$\begin{equation*} \left| f(x) - g(x) \right| = \begin{cases} f(x) - g(x) \amp \text{if $f(x) \geq g(x)$,}\\ g(x) - f(x) \amp \text{if $g(x) \geq f(x)$,} \end{cases} \end{equation*}$

to evaluate the area $A_i$ of each subregion $S_i\text{,}$ and add them up to get the area $A = A_1 + A_2 + \ldots + A_n\text{.}$

🔗

Concept 5.6.2. Vertical vs horizontal slicing.

Sometimes it is better to calculate the area of a region by slicing it with horizontal rectangles. If the region $S$ is bounded by the curves $x=f(y)\text{,}$ $x=g(y)\text{,}$ $y=c$ and $y=d\text{,}$ with $c \leq d\text{,}$ then a typical horizontal rectangle will have height $dy$ and width $|f(y) - g(y)|\text{,}$ and the area will be given by integrating from $y=c$ to $y=d\text{:}$

A = \int_{c}^{d} | f (y) - g (y) | d y .

$\begin{equation*} A = \int_c^d \left| f(y) - g(y) \right|\ dy. \end{equation*}$

If $f(y) \geq g(y)$ over the interval $y \in [c,d]$ (that is, the curve $x=f(y)$ is the right boundary of the region, while the curve $x= g(y)$ is the left boundary), then you can drop the absolute value in the expression above.

🔗

MATH 144: Calculus for the Physical Sciences I