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Notes for Math 220, Calculus 2

Brenton LeMesurier

Section 2.1 Areas between Curves

References.

Introduction.

In Section 1.1 the area below a curve and above the \(x\)-axis between \(x=a\) and \(x=b\) was described in terms of a limit of Riemann sums, and then the Fundamental Theorem of Calculus Equation (1.3.1) in Section 1.3 allowed us to avoid the sums and limits, and get the result simply as a definite integral, evaluated by finding an anti-derivative \(F(x)\) and evaluating \(F(b)-F(a)\text{.}\) This strategy will be used to solve a wide variety of problems:
  1. Approximate the solution of some problem by a sum of general form \(\sum f(x_i) \Delta x\text{,}\) with \(\Delta x\) the spacing between the successive values \(x_1,\, x_2,\, \dots\text{.}\)
  2. Get the exact solution as a limit of these approximating sums, of the form \(\displaystyle\lim_{\Delta x \to 0}\sum f(x_i) \Delta x\)
  3. Recognise this result as some definite integral \(\int_a^b f(x) \, dx\text{.}\)
  4. Evaluate if possible by finding the indefinite integral \(F(x)+C\) of \(f(x)\text{,}\) so that the result is \([F(x)]_a^b = F(b)-F(a)\text{.}\)
We start in this section with some variations on the area problem, and later see other applications like computing volumes, curve lengths, and surface areas.

Approximating the Area Between Two Curves Using Thin Rectangular Strips.

Consider the first goal of approximating the area of the region bounded above by curve \(y=f(x)\text{,}\) below by curve \(y=g(x)\text{,}\) at left by vertical line \(x=a\) and at right by \(x=b\text{.}\) We can use the following strategy to get approximations of this area.
(It is very similar to the approach used in Section 1.1 to approximate the area under a curve and we use a similar strategy several more times in this course, so I strongly recommend that you make sketches illustrating the strategy as we go along.)
  1. First, we can slice the region into \(n\) thin vertical strips all of equal width \(\Delta x = (b-a)/n\text{,}\) with the cuts being the vertical lines \(x=x_i\text{,}\) \(x_i=a+ih\text{,}\) \(1 \leq i \leq n-1\text{.}\) (The extremities are at \(x_0=a\) and \(x_n=b\text{.}\)) Thus strip 1 is the interval \([x_0, x_1]\text{,}\) ending at \(x_1\text{,}\) and so on: strip number \(i\) is \([x_{i-1}, x_i]\text{.}\)
  2. The area of each strip can be approximated by that of a rectangle roughly fitting the strip: width \(\Delta x\) and height the vertical distance between the curves at some point within the strip.
  3. Where do we measure the height of the \(i\)-th rectangle? At any \(x\) value \(x_i^*\) within the strip, so \(x_{i-1} \leq x_i^* \leq x_i\text{.}\) (One natural choice is the midpoints, \(x_i^* = (x_{i-1} + x_i)/2\text{.}\))
  4. The \(i\)-th rectangular strip extends vertically from \(y=g(x_i^*)\) to \(y=f(x_i^*)\) and so has height \(f(x_i^*)-g(x_i^*)\) and area
    \begin{equation*} (f(x_i^*)-g(x_i^*)) \Delta x. \end{equation*}
  5. The total area of the region between the curves is approximated by the sum of these strip areas:
    \begin{equation*} A_n = \sum_{i=1}^n (f(x_i^*)-g(x_i^*)) \Delta x. \end{equation*}

The Exact Area Between Two Curves.

We can improve the above approximation \(A_n\) by increasing the number of strips \(n\) and so decreasing their width \(\Delta x\text{,}\) and it is reasonable that the exact area is the limit as \(n \to \infty\text{:}\)
\begin{equation} A = \lim_{n \to \infty} A_n = \lim_{n \to \infty} \sum_{i=1}^n (f(x_i^*)-g(x_i^*)) \Delta x.\tag{2.1.1} \end{equation}
Now the good news, critical to a lot of what we do in this course: we do not need to do anything with this limit and sums except compare this formula
\begin{equation*} A = \lim_{n \to \infty} \sum_{i=1}^n (f(x_i^*)-g(x_i^*)) \Delta x \end{equation*}
to the definition of the definite integral
\begin{equation*} \int_a^b f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x \end{equation*}
(see Equation (1.2.1) in Section 1.2) with the same meaning for the \(x_i\text{,}\) \(x_i^*\) and so on.
This comparison gives the area of this region between the curves and vertical lines as
\begin{equation} A = \int_a^b f(x)-g(x) \, dx\tag{2.1.2} \end{equation}
  • Sketch the region bounded above by \(y=e^x\text{,}\) bounded below by \(y=x\text{,}\) and bounded at the sides by \(x=0\) and \(x=1\text{.}\)
  • Find its area.

Regions Bounded by Intersections of Curves.

Sometimes a region is naturally enclosed by two curves \(y=f(x)\text{,}\) \(y=g(x)\) that intersect, and the left and right extremities come from those intersections, rather than from explicitly given \(x\) values. Then the limits of integration come from solving \(f(x)=g(x)\) to find the “corners”.
  • Sketch the region enclosed by the parabolas \(y=x^2\) and \(y=2x-x^2\)
  • Find its area.

Which is the Top, Which is the Bottom?

Sometimes it is is not initially obvious which function describes the top and which the bottom, and instead you have to work this out. Also, the curves can swap places if they intersect repeatedly. The safe way is to note that the height of the region at any \(x\) value is always \(|f(x)-g(x)|\text{,}\) so the area is
\begin{equation*} A = \int_a^b | f(x)-g(x) | \, dx \end{equation*}
However, to find anti-derivatives it helps to have a formula that does not involve absolute values. The way to do that is work out the intervals where \(f(x)>g(x)\) giving height \(f(x)-g(x)\) and those where the opposite is true so the height is \(g(x)-f(x)\text{;}\) then integrate these different height functions over each such interval, and add the results. Only one anti-derivative need by found: just flip its sign! In turn, the way to find the \(x\) values that divide these intervals is to note that they are the intersections of the curves, found by solving \(f(x)=g(x)\text{:}\) a sketch can then help to work out the top and bottom in each sub-interval. (Inequalities are often best handled like this, by first solving for equality.)
  • Sketch the region enclosed by the curves \(y=\sin x\text{,}\) \(y=\cos x\text{,}\) \(x=0\) and \(x=\pi/2\)
  • Find its area.

Rotating and Chopping to Simplify, and Geometry.

Some regions are not best described by top and bottom curves as functions of \(x\text{.}\) Instead, the best approach might be to look at things sideways, with curves at left and right and using \(y\) as the variable of integration. Another strategy is to divide the region into several simpler pieces and add their areas.
(And sometimes, basic geometry is the better way to compute an area!)
  • Sketch the region enclosed by the line \(y=x-1\) and the parabola \(y^2=2x+6\text{.}\)
  • Then try several different strategies for computing its area.
openstax.org/books/calculus-volume-2/pages/2-1-areas-between-curves
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