Evaluating integrals geometrically, without using the fundamental theorem of calculus

Question

I'm designing a lesson for an Introduction to Integral Calculus class, and I want to encourage students to evaluate integrals without just going straight for the antiderivative and using the fundamental theorem of calculus. I want them to think geometrically about the situation before diving in with computations. Here are a few general topics I've thought to include so far:

• Integration of even and odd functions.

• Integration of functions with graphs that are familiar shapes. Like $\int_{-2}^2 \sqrt{4-x^2}\;\mathrm{d}x$.

• Maybe integrating a function without an nice antiderivative on the domain of integration, like $\int_{-a}^a |x|\;\mathrm{d}x\,$? But I would want the function so be more interesting that just $|x|$.

What are good examples of integrals that illustrate the ideas above and would work well in a lesson? Or better, does anyone have other ideas for topics that I should (or shouldn't) include in this lesson?

Answer 1

A useful trick is the idea of rearranging values of a function. For example, while $\sin^2 \theta$ and $\cos^2 \theta$ have different graphs on $0 \leq \theta \leq \pi/2$ you can tell from the graphs of $\sin \theta$ and $\cos \theta$ that the values which the squared functions take must be equal. In short, $$ \int_{0}^{\pi/2} \cos^2 \theta \, d\theta = \int_{0}^{\pi/2} \sin^2 \theta \, d\theta $$ Of course, you can make the same conclusion for other intervals such as $[0, \pi]$ or any $2\pi$-length interval. Use the Pythagorean identity and the fact $\int_{a}^{b} c \, d\theta = c(b-a)$ to derive: $$ \frac{\pi}{2} = \int_{0}^{\pi/2} 1 \, d\theta = \int_{0}^{\pi/2} \left(\cos^2 \theta +\sin^2 \theta\right) \, d\theta = \int_{0}^{\pi/2} \cos^2 \theta \, d\theta + \int_{0}^{\pi/2} \sin^2 \theta \, d\theta $$ Hence, $\displaystyle \int_{0}^{\pi/2} \cos^2 \theta \, d\theta = \int_{0}^{\pi/2} \sin^2 \theta \, d\theta = \pi/4$.

This trick paired with those which exploit the fact that multiples of a period of a sinusoid integrate to zero can save a lot of work on common trigonometric integrals. More to the point, these are relevant to your lesson because they allow integration without a deep understanding of yet-taught techniques

( I assume students are largely ignorant of basic trigonometric identities like $\sin^2 \theta = \frac{1}{2} \left( 1 - \cos 2 \theta \right)$, a belief those outside the USA will likely gasp in disbelief, but those within the US will easily recognize as a realist position given current malpractice in our standards of precalculus etc.)

Answer 2

In his Calculus book, Spivak gives these two exercises before he ever introduces the fundamental theorems of calculus.

Evaluate without doing any computations:

$$\int\limits_{-1}^{1} x^3\sqrt{1-x^2} \,\mathrm{d}x \qquad\qquad\int\limits_{-1}^{1} \left(x^5+3\right)\sqrt{1-x^2} \,\mathrm{d}x$$

Answer 3

The integrals of sine and cosine have a nice geometric interpretation, if you and your students are comfortable treating the differential of a circular arc as equivalent to a straight hypotenuse (in the limit as $\Delta\theta \rightarrow 0$). This is perhaps more of an example than a simple exercise for the student. But I think it answers the OP's objective of thinking geometrically about integrals instead of in terms of algebraic antidifferentiation.

For instance, $\int_a^b \sin\theta \; d\theta = \cos a - \cos b$:

One may find the reasoning (which I assume the reader can fill in) a little loose, but it's a true picture that helps give the student a new perspective on sine and cosine. I prove the ratio of the lengths of an arc and its subtending chord approaches $1$ as the lengths approach zero as well as prove the derivatives in a similar geometric fashion, so it feels more comfortable to me.

This idea may be found in Pascal, Traité des sinus du quart de cercle 1659, Prop. I: La somme des sinus d’un arc quelconque du quart de cercle est égale à la portion de la base comprise entre les sinus extrêmes multipliée par le rayon.

Answer 4

Notice through a geometric argument that

$$\int_0^af(x)\ dx=\int_0^af(a-x)\ dx$$

$$\int_0^af(x)\ dx=\frac1b\int_0^{ab}f(x/b)\ dx$$

The second which may follow through a squeezing of the integrals or Riemann sums.

Now compute the following integrals:

$$\int_0^{\pi/2}\frac{\sin^\pi(\theta)}{\sin^\pi(\theta)+\cos^\pi(\theta)}\ d\theta$$

$$\int_0^{\pi/2}\ln(\sin(\theta))\ d\theta$$

For the first integral:

\begin{align}\int_0^{\pi/2} \frac{\sin^\pi(\theta)}{\sin^\pi(\theta)+\cos^\pi( \theta)}\ d\theta &=\int_0^{\pi/2}\frac{\sin ^\pi( \frac\pi2-\theta)}{\sin^\pi(\frac\pi2-\theta) +\cos^\pi(\frac\pi2-\theta)}\ d\theta\\& =\int_0^{\pi/2}\frac{\cos^\pi( \theta)}{\sin^\pi(\theta)+\cos^\pi( \theta)}\ d\theta\\ \therefore I+I=\int_0^{\pi/2} \frac{\sin^\pi(\theta) + \cos^\pi(\theta)}{\sin^\pi(\theta)+\cos^\pi( \theta)}\ d\theta&=\int_0^{\pi/2}1\ d\theta = \frac\pi2\\\therefore\int_0^{\pi/2} \frac{\sin^\pi(\theta)}{\sin^\pi(\theta)+\cos^\pi( \theta)}\ d\theta &=\frac\pi4\end{align}

For the second integral:

\begin{align} \int_0^{\pi /2} \ln(\sin(\theta))\ d\theta = \int_0^{\pi /2} \ln(\cos(\theta))\ d\theta\end{align} \begin{align} \therefore I+I=\int_0^{\pi /2}\ln(\sin(\theta))+\ln(\cos(\theta)) \ d\theta &= \int_0^{\pi/2} -\ln(2)+\ln(2\sin(\theta)\cos(\theta)) \ d\theta \\&= -\frac\pi2\ln(2) +\frac12\int_0^\pi \ln(\sin(\theta))\ d\theta \\&= -\frac\pi2\ln(2) +I\end{align} \begin{align}\therefore \int_0^{\pi /2} \ln(\sin(\theta))\ d\theta =-\frac\pi2\ln(2)\end{align}

Answer 5

There are many nice problems involving integrals of the "integer floor" function, which are evaluated as series. Eg questions 2 and 3 here: https://maths.org/step/sites/maths.org.step/files/assignments/assignment3.pdf

Answer 6

Just ask the students to think geometrically about the basic properties of integrals.

Thinking of an integral as "the area under the curve," write a geometric explanation of why each of the following properties of integrals is reasonable to believe. $$\int\limits_a^b cf(x) \,\mathrm{d}x \;=\; c\!\int\limits_a^b f(x) \,\mathrm{d}x \qquad\qquad \int\limits_a^b f(x) \,\mathrm{d}x \;=\; \int\limits_0^b f(x) \,\mathrm{d}x - \!\int\limits_0^a f(x) \,\mathrm{d}x\\ \int\limits_a^b \big(f(x)+g(x)\big) \,\mathrm{d}x \;=\; \int\limits_a^b f(x) \,\mathrm{d}x + \!\int\limits_a^b g(x) \,\mathrm{d}x$$

Or for more of a challenge, ask the same question for the property $$ \int\limits_0^a f(x) \,\mathrm{d}x = \int\limits_0^a f(a-x) \,\mathrm{d}x $$ If they've covered $u$-substitution it would be reasonable to ask them to prove this formally.

Answer 7

Based on a discovery of Gregory of St. Vincent, Opus geometricum quadraturae circuli et sectionum coni (1647), Props. CVIII-CIX, 585-586:

Consider integrating the hyperbola $y = 1/x$:

If the $x$ coordinates of a partition of the interval from $1$ to $b=2$ are multiplied by $a=3$, the partition is transformed to one of the interval from $a = 3$ to $ab=6$. Furthermore, all the rectangles in a Riemann sum over $[1,2]$ are transformed to equal rectangles in a Riemann sum over $[3,6]$, since the bases are multiplied by $a=3$ and the heights multiplied by $1/a=1/3$.

1. Why does this show $\int_1^2 {dx \over x} = \int_3^6 {dx \over x}$? Argue similarly that $\int_1^b {dx \over x} = \int_a^{ab} {dx \over x}$ for any $a,b>0$).

2. Let $L(a) = \int_1^a {dx \over x}$. Show that $L(ab)=L(a)+L(b)$. Such a function is called a logarithmic function. (First argue that $\int_1^{ab} {dx \over x} = \int_1^{a} {dx \over x} + \int_a^{ab} {dx \over x}$.)

3. Given that $L(a)$ is a logarithm, estimate its base. (First recall what the value of $\log_B(B)$ is.)

Answer 8

From

we have

$$\int_0^x \sqrt{a^2-t^2}\;dt = {1 \over 2}\,x\,\sqrt{a^2-x^2} + {1\over2}\,a^2 \arcsin\left({x \over a}\right)$$

Exercise: Show that the derivative of the right-hand side is $\sqrt{a^2-x^2}$.

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Similarly, from

we have

$$\int_0^t \arcsin x \; dx = t\,a - \int_0^a \sin y \; dy$$

where $a = \arcsin t$.

This one may be best kept for when the students know how to integrate $\sin y$.

Answer 9

I like the explication and just the example in this paper (really makes you think about an accumulation and a rate). Very nice graphics also (for box counting).

http://home.hiroshima-u.ac.jp/yhiraya/er/ZR11_Z_03.html

Note, he ended up underestimating hydrocarbon production but the thought process is still very nice. And I think doing something topical like this makes it relevant. Not just another cannonball.