Calculus II

Calculus??

We might think of Calculus as that part of mathematics that is dependent on the notion of limit.

Calculus I??

In Calculus I, the main objects of study are derivatives and integrals - though, you probably only just started integration.

I've got a little table that outlines the parallels between the development of differential calculus and that of integral calculus:

So, what's next?

• Techiques of integration
• Applications
• Sequences and series

And, what do you need to know about integration?

• The geometric interpretation
• The definition
• The basics of anti-differentiation
• The fundamental theorem of calculus

Interpretation of the integral

You should know that definite integral can be interpreted as signed area. Roughly, $$\begin{align} \int_a^b f(x) \, dx & = \text{the area under the graph and over the } x \text{ axis} \\ & - \text{the area over the graph and under the } x \text{ axis.}\end{align}$$

A sample problem

The complete graph of a function $f$ is shown below; it consists of a quarter circle and a straight line segment. Compute $$\int_{-2}^3 f(x) \, dx.$$

Solution

The green portion above the $x$-axis consists of a quarter of a circle of radius 2 (so that it's area is $\frac{1}{4}\pi \times 2^2 = \pi$) and a triangle whose area is $2$. The red portion below the $x$-axis consists of a single triangle whose area is $1/2$. Thus, $$\int_{-2}^3 f(x) \, dx = \pi + 2 - \frac{1}{2}.$$

The definition of the integral

$$ \begin{aligned} \int_a^b f(x) \, dx &= \lim_{n\to\infty} \sum_{i=1}^n f(x_i) \, \Delta x \: \text{ (or)}\\ & = \lim_{n\to\infty} \sum_{i=1}^n f(x_{i-1}) \, \Delta x, \end{aligned} $$ where $\Delta x = \frac{b-a}{n}$ and $x_i = a+i\Delta x$.

Use the second version (with the $x_{i-1}$) when you want a left sum and use the first version (with the $x_i$) when you want a right sum.

The sum and area

These sums literally mean - please add up the areas of the rectangles. $$ \text{Left: }\sum_{i=1}^n f(x_{i-1}) \, \Delta x \: \: \: \: \: \: \: \: \text{ Right: } \sum_{i=1}^n f(x_{i}) \, \Delta x $$

A sample problem

Estimate $$ \int_0^2 (2 + 4 x - x^2) \, dx$$ using a left Riemann sum with $n=4$ terms.

Solution 1

We break the interval $[0,2]$ into $n=4$ pieces so $$\Delta x = \frac{b-a}{n} = \frac{2-0}{4} = \frac{1}{2}.$$ Thus, $x_i = a + i\,\Delta x = 0 + i\times\frac{1}{2} = i/2$ and $$ \int_0^2 (2 + 4 x - x^2) \, dx \approx \sum_{i=1}^4 \left(2 + 4 \left(\frac{i-1}{2}\right) - \left(\frac{i-1}{2}\right)^2\right)\,\frac{1}{2} = 8.25.$$ You can compute this type sum using Sage very easily.

Solution 2

If you can visualize the sum, it's pretty easy to see that the points you need to plug in are $0$, $0.5$, $1$, and $1.5$.

Thus, $$ \begin{align}\int_0^2 (2 + 4 x - x^2) \, dx & \approx (2+0+0)\times0.5 \\ &+ (2 + 4\times0.5 - 0.5^2)\times0.5 \\ &+ (2 + 4\times1 - 1^2)\times0.5 \\ &+ (2 + 4\times1.5 - 1.5^2)\times0.5 \\ &= 8.25.\end{align}$$

Basic definite and indefinte integrals using FTC

The fundamental theorem of calculus states that $$\int_a^b f(x) \, dx = F(b) - F(a),$$ provided that $F'=f$. Thus, we can evaluate definite integrals by finding anti-derivatives. Because of this close, inverse connection between integration and differentiation, we define the so-called indefinite integral as a convenient notation to represent and anti-deriviate, i.e. $$ \int f(x) \, dx = F(x) + c, \text{ where } F' = f. $$

Sample indefinite integral

$$\int (x^2 + x + 1) \, dx = ?$$

Sample definite integral

$$\int_0^2 (2x - x^2) \, dx = ?$$

Techniques of integration

Don't let the simplicity of the previous examples lull you into complacency. Integration is quantifiably more difficult than differentiation, as this demonstration concretely illustrates.

Over the next couple of weeks, we'll carefully discuss

• $u$-substitution,
• Numerical integration, and
• Integration by parts.