Welcome back to Professor Baker's Math Class! In this set of notes covering Chapter 5-2 through 5-4, we transition from approximating areas with Riemann sums to calculating them exactly using algebraic techniques. This is a pivotal moment in calculus where differentiation and integration are officially connected.

1. Properties of the Definite Integral

Before we dive into solving complex problems, we established some ground rules for definite integrals. These properties allow us to manipulate integrals to make them easier to solve:

  • Reversing Limits: If you swap the bounds of integration, the sign changes: $$\int_b^a f(x) dx = -\int_a^b f(x) dx$$
  • Zero Width Interval: Integrating from a point to itself results in zero area: $$\int_a^a f(x) dx = 0$$
  • Linearity: You can split sums and pull out constants: $$\int_a^b [f(x) \pm g(x)] dx = \int_a^b f(x) dx \pm \int_a^b g(x) dx$$

2. The Fundamental Theorem of Calculus (FTC)

This theorem is the bridge between derivatives and integrals. It has two parts:

Part 1: The Derivative of an Integral

If $g(x) = \int_a^x f(t) dt$, then $g'(x) = f(x)$. Essentially, differentiation undoes integration.

Watch out for the Chain Rule! If the upper limit is a function of $x$ rather than just $x$, you must multiply by the derivative of that limit. From our class notes:

$$ \frac{d}{dx} \int_1^{x^4} \sec t \, dt = \sec(x^4) \cdot \frac{d}{dx}(x^4) = \sec(x^4) \cdot 4x^3 $$

Part 2: The Evaluation Theorem

This provides the method for evaluating definite integrals without using Riemann sums:

$$ \int_a^b f(x) dx = F(b) - F(a) $$

where $F$ is any antiderivative of $f$. For example:

$$ \int_1^3 e^x dx = e^3 - e^1 $$

3. Indefinite Integrals

When there are no limits of integration, we are finding a family of functions. Never forget to add the constant of integration, $+ C$.

Example from the notes:

$$ \int (10x^4 - 2\sec^2 x) dx = \frac{10x^5}{5} - 2\tan x + C = 2x^5 - 2\tan x + C $$

4. Application: Displacement vs. Total Distance

One of the most common applications of integrals is particle motion. Given a velocity function $v(t)$, there is a distinct difference between displacement and distance:

  • Displacement (Net Change): Simply integrate velocity over the time interval. $\int_{t_1}^{t_2} v(t) dt$.
  • Total Distance Traveled: You must integrate the absolute value of velocity. $\int_{t_1}^{t_2} |v(t)| dt$.

Strategy: To find total distance, determine where $v(t) = 0$. If the particle changes direction within your time interval (like at $t=3$ in our class example), you must split the integral into two parts and add the absolute values of the areas.

5. A Critical Warning on Continuity

As discussed in the "What is wrong with this calculation?" slide, you cannot apply the Fundamental Theorem of Calculus if the function is discontinuous within the interval $[a, b]$. For example, $\int_{-1}^3 \frac{1}{x^2} dx$ cannot be evaluated directly because the function is undefined at $x=0$. Always check your domain!

Keep practicing your antiderivative rules, and remember to check your signs when evaluating $F(b) - F(a)$. See you in the next lecture!