What is a definite integral?
A derivative measures a rate at one instant. A definite integral totals a rate across an interval.
Add up the area under y = 2x
From x = 0 to x = 3, the line y = 2x rises from height 0 to height 6. The region beneath it is a triangle with base 3 and height 6.
Area = ½ × 3 × 6 = 9, so ∫03 2x dx = 9.
You can check the same result with an antiderivative: the derivative of x² is 2x, so the Fundamental Theorem of Calculus gives 3² − 0² = 9.
Why “signed” area matters
If the graph falls below the x-axis, those pieces count negatively in a definite integral. When the graph stays above the axis, as it does here, the integral is ordinary positive area. Integrals also model accumulated distance, growth, and many other totals when the function is a rate. See how a derivative describes the rate itself.
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