Welcome back, class! In Chapter 5, we arrive at a pivotal moment in our Calculus journey. After spending the first half of the course breaking functions down with derivatives (finding slopes), we are now learning how to build them back up using Integrals (finding areas). This is the reverse process of differentiation, often called finding the antiderivative.
1. The Area Problem and Antiderivatives
We started by looking at the geometry of graphs. As shown in the notes, if we have a constant rate of change (a horizontal line $f'(x) = 3$), the area under that line forms a rectangle ($A=lw$). If we have a linear function like $y=2x$, the area forms a triangle ($A=\frac{1}{2}bh$).
This geometric concept leads us to the notation of the integral. If we know the derivative $f'(x)$, we can find the original function $f(x)$:
- Differentiation: $f(x) \rightarrow f'(x)$
- Integration: $f'(x) \rightarrow \int f(x) dx$
2. The Fundamental Theorem of Calculus (FTC)
The bridge between Reimann sums (approximating area with rectangles) and exact area is built on the Fundamental Theorem of Calculus. As highlighted on Page 15 of your notes, this theorem allows us to evaluate definite integrals by plugging in the bounds:
$$ \int_a^b f(x) dx = F(b) - F(a) $$
Where $F$ is the antiderivative of $f$. This means to find the area under the curve from $x=a$ to $x=b$, we integrate the function and subtract the value at the lower limit from the value at the upper limit.
3. Essential Integration Rules
Just as we had rules for derivatives, we have rules for integrals (see Page 17). You should be comfortable with:
- Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (where $n \neq -1$)
- Exponential Rule: $\int e^x dx = e^x + C$
- Trigonometric Integrals: Knowing that $\int \cos x dx = \sin x + C$ and $\int \sec^2 x dx = \tan x + C$.
4. The Chain Rule in Reverse: U-Substitution
The latter half of this chapter introduces one of the most important tools in your integration toolkit: U-Substitution. This technique allows us to integrate composite functions by changing the variable from $x$ to $u$.
The Strategy:
- Choose $u$ to be the "inside" function (often inside a square root, parenthesis, or denominator).
- Find the derivative $du$.
- Match the parts of the integral to substitute $dx$ with terms of $du$.
- Integrate with respect to $u$.
- Substitute back to $x$ (for indefinite integrals).
Example from class notes:
To solve $\int 2x\sqrt{1+x^2} dx$, we set $u = 1+x^2$. Consequently, $du = 2x dx$. The integral simplifies perfectly to $\int \sqrt{u} du$, which is much easier to solve!
Keep practicing those U-Subs! Identifying the correct $u$ takes practice, but once you see the pattern, it becomes second nature. Good luck with your studies!