Overview
This lecture introduces the concept of integration in calculus, explaining how to recover a function from its derivative (slope) and connecting integration to the area under a curve.
From Slope to Height (Integration)
- Calculus involves two related functions: a height function (y(x)) and its slope (s(x)).
- Previously, we found the slope (derivative) from the height; now we go backwards to find height from slope.
- For power functions, if y = xⁿ, then dy/dx = n*xⁿ⁻¹; to reverse, integrate xⁿ to get xⁿ⁺¹/(n+1).
- Many slopes fit basic forms (powers, sines, cosines, eˣ, logx), which helps in reverse-engineering their heights.
The Summation and the Integral
- The process starts with discrete steps: adding up all small changes (delta y) gives the total change in y.
- The sum of the increments (slopes times small intervals) approximates the total height gained.
- As the intervals become infinitesimally small (delta x → 0), the sum becomes an integral.
- The definite integral of s(x) from a to b equals y(b) - y(a), representing the net change in height.
The Symbol and Meaning of the Integral
- The integral symbol ∫ s(x) dx is used for the process of finding the height from the slope.
- The integral generalizes the sum of small products (slope × width) as intervals shrink towards zero.
Example: Integrating a Linear Slope
- For s(x) = 2 – 2x, integrating involves splitting the x-range into small pieces, approximating the area with rectangles.
- As rectangles get thinner, the sum approaches the true area under the curve.
- The integrated function y(x) is found as y = 2x – x² because its derivative matches s(x).
- At x = 1, the area under s(x) is 1; at x = 1/2, it is 3/4; the area matches the value of the integrated function at those points.
- Negative area below the x-axis reflects a decrease in height or reversing direction.
Key Terms & Definitions
- Derivative (dy/dx) — the slope of a function at a point.
- Integral (∫ s(x) dx) — the process of finding the original function (height) from its derivative (slope).
- Definite Integral — calculates the net area under s(x) between two points.
- Infinitesimal (dx) — an indefinitely small change in x, used in integrals.
- Area Under the Curve — graphical interpretation of the integral.
Action Items / Next Steps
- Practice integrating various simple functions, especially power, exponential, and trigonometric functions.
- Review the process of summing small increments to approximate an area before taking the limit.
- Prepare for further examples and applications in upcoming lectures.