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Imagine trying to calculate the exact area under a swooping curved graph by slicing it into millions of tiny rectangles.
Adding infinite rectangles sounds impossible, but calculus reveals an astonishing shortcut: finding area is the exact reverse of finding speed or slope.
A definite integral represents the net signed area bounded between a curve y=f(x)y = f(x), the xx-axis, and two vertical boundaries x=ax = a and x=bx = b.
How do we compute this shaded area precisely without ever adding up rectangles?

The Fundamental Theorem of Calculus

In the late 1600s, Isaac Newton and Gottfried Wilhelm Leibniz independently discovered that differentiation and integration are inverse operations.
The Fundamental Theorem of Calculus (FTC) states that if f(x)f(x) is continuous on [a,b][a, b] and F(x)F(x) is any antiderivative of f(x)f(x) such that Fβ€²(x)=f(x)F'(x) = f(x), then: ∫abf(x) dx=F(b)βˆ’F(a)\int_a^b f(x)\,dx = F(b) - F(a)
To evaluate this on paper, we write the antiderivative inside square brackets with the limits on the right: [F(x)]ab=F(b)βˆ’F(a)\left[F(x)\right]_a^b = F(b) - F(a).
Before evaluating an integral, let's refresh how to reverse a power: by the power rule for integration, the antiderivative of xnx^n is xn+1n+1\frac{x^{n+1}}{n+1}.
What does this look like with a full polynomial?

Worked Example: Evaluating a Definite Integral