Introduction to Integrals

Your first step into calculus — measuring areas, distances, and totals

You already know how to find the area of a rectangle: length × width. But what about the area under a curve? That's the big question that calculus was invented to answer — and it's called integration.

The trick: fill the curved area with rectangles you can measure, and add them up. The more rectangles, the closer to the true area. This idea — the Riemann sum — is how integrals work under the hood.

In this lesson, you'll build rectangles under y = 4 - x^2, watch the approximation improve as you add more, and discover why integrals matter everywhere — from physics (distance from speed) to economics (total revenue from a rate).

Related Topics

What is a Derivative?Quadratic EquationsPower Functions

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What is an integral?
An integral measures the total area under a curve between two points. \int_a^b f(x)\,dx gives the net signed area.
What is a Riemann sum?
A Riemann sum approximates the area by dividing the interval into n rectangles and summing their areas. Width = (b−a)/n, height = f(x*) at some sample point.
How does the number of rectangles affect accuracy?
More rectangles = better approximation. With 4 rectangles: ≈ 5.375. With many rectangles: approaches the exact value 16/3 ≈ 5.333.
What is the Fundamental Theorem of Calculus?
Integration and differentiation are inverse operations. If F'(x) = f(x), then \int_a^b f(x)\,dx = F(b) - F(a).
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