If Differential Calculus is about breaking things down to find an instantaneous rate of change (like speed), Integral Calculus is about putting them back together to find a total (like distance traveled). It is often called "Anti-differentiation".
1. The Big Idea: Reverse Engineering
Differentiation tells you the slope. Integration takes the slope and tells you the original curve.
You have x². The derivative is 2x.
Integration:
You have 2x. The integral is x²... (plus a constant).
2. Indefinite Integrals and the "+ C"
When we find the derivative of x², we get 2x.
When we find the derivative of x² + 10, we also get 2x (because the derivative of a constant 10 is zero).
So, if we start with 2x and go backward, we don't know if the original function was x², x² + 5, or x² - 100. To account for this unknown starting value, we add "C" (the Constant of Integration).
[Image of family of curves for indefinite integral]3. The Definite Integral: Area Under the Curve
While the indefinite integral gives us a function, the Definite Integral gives us a number. It calculates the exact area under a curve between two specific points, a and b.
[Image of area under curve integral]∫ (from a to b) f(x) dx
Imagine slicing the area under a graph into infinite tiny rectangles and adding them all up. That is integration.
4. The Fundamental Theorem of Calculus
This theorem connects the two branches of calculus (derivatives and integrals). It states that if you want to find the area under f(x) from a to b:
- Find the antiderivative, let's call it F(x).
- Calculate F(b) - F(a).
Antiderivative of 2x is x².
Evaluate at 3: (3)² = 9
Evaluate at 1: (1)² = 1
Result: 9 - 1 = 8
5. Real World Applications
Why do we need to accumulate things?
- Physics: Integrating velocity gives you displacement (distance). Integrating acceleration gives you velocity.
- Engineering: Calculating the center of mass or the total force on a dam wall.
- Geometry: Finding the volume of complex 3D shapes (like vases or domes) by "rotating" a curve.