Rain filling a bucket
Rain falls at a changing rate all afternoon. To find the total water collected, you add up tiny slivers — a little in each second — into one sum. The integral is that adding-up of infinitely many thin slices into a whole.
Integration accumulates
Where the derivative measures a rate of change, the integral accumulates. The definite integral of a function from a to b equals the signed area between its graph and the x-axis over that interval — imagined as summing infinitely many thin rectangles.
The Fundamental Theorem
Integration and differentiation are inverses. To integrate, find an antiderivative F and evaluate F(b) − F(a). The power rule reverses: ∫xⁿ dx = xⁿ⁺¹/(n+1).
∫₀² 2x dx: an antiderivative of 2x is x². Evaluate 2² − 0² = 4 — the area under y = 2x from 0 to 2.
Lab · Compute an area
- Take the definite integral of 3x² from 0 to 2.
- Find an antiderivative (reverse the power rule).
- Evaluate it at 2 and subtract its value at 0.
What you should see: An antiderivative of 3x² is x³. Then 2³ − 0³ = 8 — the area under the curve.