Welcome to the second major pillar of Calculus! Having mastered derivatives (the study of rates of change), we now shift our focus to Integrals (the study of accumulation and area). This week, we cover Sections 4.9 and 5.1, which lay the foundation for the Fundamental Theorem of Calculus.

Section 4.9: Antiderivatives

In this section, we learn to reverse the process of differentiation. If you are given a function $f(x)$, the goal is to find a function $F(x)$ such that $F'(x) = f(x)$. This function $F$ is called an antiderivative.

Key Concepts:

  • The General Antiderivative: Because the derivative of a constant is zero, there is a family of antiderivatives for any function. We express this by adding an arbitrary constant, $C$.
    $$ \int f(x) \, dx = F(x) + C $$
  • The Power Rule for Integrals: Just as we subtracted from the exponent for derivatives, we add to it for integrals: $$ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad (n \neq -1) $$
  • Applications: We examine rectilinear motion. If given an acceleration function $a(t)$, we can antidifferentiate to find velocity $v(t)$, and antidifferentiate again to find position $s(t)$.

Section 5.1: Areas and Distances

While Section 4.9 focuses on the algebra of integrals, Section 5.1 focuses on the geometry. This section introduces the Area Problem: How do we find the exact area of the region that lies under the curve $y = f(x)$?

Key Concepts:

  • Approximation with Rectangles: We cannot strictly measure the area of a curved region using basic geometry, so we approximate it using rectangles. The width of each rectangle is usually denoted as $\Delta x$.
  • Sample Points: We determine the height of our rectangles using specific points on the curve:
    • Right Endpoints ($R_n$): Using the right side of the sub-interval.
    • Left Endpoints ($L_n$): Using the left side of the sub-interval.
    • Midpoints ($M_n$): Using the center of the sub-interval.
  • Distance as Area: We learn that the area under a velocity-time graph represents the total distance traveled by an object.
  • Limit Definition: We begin to see that as the number of rectangles ($n$) approaches infinity, the approximation becomes the exact area: $$ A = \lim_{n \to \infty} R_n = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x $$

Please review the PowerPoint slides and lecture notes below before attempting the homework. Pay special attention to the algebraic manipulation in 4.9—forgetting the "$+ C$" is the most common mistake students make!

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