Welcome back to Professor Baker's Math Class! In this set of notes, we are making a monumental shift in Calculus 1. We have spent the semester mastering derivatives (slopes of tangent lines); now, we move to integrals (areas under curves). This post covers the crucial concepts found in Sections 5.1 through 5.4, taking you from geometric estimations to precise calculations.

1. From Approximations to Exact Areas (5.1 - 5.2)

How do we find the area under a curve that isn't a simple rectangle or triangle? We start by approximating. As shown in the notes, we can estimate the area under a curve like $y=x^2$ using rectangles. This is known as a Riemann Sum.

By using left endpoints or right endpoints for the height of our rectangles, we get an estimate. However, to get the exact area, we need to take the limit as the number of rectangles ($n$) goes to infinity:

$$\text{Area} = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x = \int_{a}^{b} f(x) \, dx$$

This limit definition gives birth to the Definite Integral. The symbol $\int$ (an elongated 'S' for sum) represents the accumulation of area between the curve $f(x)$ and the x-axis from $x=a$ to $x=b$.

2. The Fundamental Theorem of Calculus (5.3)

Calculating limits of sums is tedious. Fortunately, Section 5.3 introduces the Fundamental Theorem of Calculus (FTC), which establishes the connection between differential calculus and integral calculus. It essentially states that differentiation and integration are inverse processes.

FTC Part 2 provides the practical method we use for evaluation:

$$\int_{a}^{b} f(x) \, dx = F(b) - F(a)$$

Where $F$ is any antiderivative of $f$ (meaning $F' = f$).

Example from class notes:
To find the area under $y=x^2$ from 0 to 2:

$$\int_{0}^{2} x^2 \, dx = \left[ \frac{1}{3}x^3 \right]_{0}^{2} = \frac{1}{3}(2)^3 - \frac{1}{3}(0)^3 = \frac{8}{3}$$

3. Indefinite Integrals and Anti-Differentiation Formulas (5.4)

When we find a general antiderivative without specific limits ($a$ and $b$), we call it an Indefinite Integral. The result is a function plus a constant of integration, $C$.

Here are some essential integration rules derived from our derivative rules (Power Rule, Trig, and Exponentials):

  • Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ (where $n \neq -1$)
  • Exponential: $\int e^x \, dx = e^x + C$
  • Natural Log: $\int \frac{1}{x} \, dx = \ln|x| + C$
  • Trigonometry:
    • $\int \cos x \, dx = \sin x + C$
    • $\int \sec^2 x \, dx = \tan x + C$
    • $\int \csc x \cot x \, dx = -\csc x + C$

4. Net Change: Displacement vs. Distance

An important application of integrals is motion. If we are given a velocity function $v(t)$, integration allows us to find position concepts.

  • Displacement: The net change in position is simply $\int_{a}^{b} v(t) \, dx$.
  • Total Distance Traveled: This requires us to integrate the absolute value of velocity (speed), or split the integral at points where the object changes direction (where $v(t)=0$).

Key Takeaway

Remember the properties of integrals shown in the notes: you can split integrals over intervals ($ \int_a^c + \int_c^b $), and pull out constant multiples. Practice using the table of integrals provided in Section 5.4, as memorizing these antiderivatives is key to success on the upcoming exam.

Happy studying, and see you in the next lecture!