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Pure · Integration

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Definite integration and area under a curve

Why evaluating F(b) − F(a) gives the area under a curve, what a negative integral actually means, and why "integral" and "area" are not always the same number.

Pure · Integration

Definite integration and area under a curve

Why evaluating F(b) − F(a) gives the area under a curve, what a negative integral actually means, and why "integral" and "area" are not always the same number.

Why it works

Here's the idea that makes definite integration click. Imagine sweeping from left to right under the curve y=f(x)y = f(x), and let A(x)A(x) be the area collected so far. How fast is that area growing? When you move a tiny step dxdx to the right, you add a sliver that is dxdx wide and (almost exactly) f(x)f(x) tall — so you gain about f(x)⋅dxf(x) \cdot dx of area. Gain per unit step:

dAdx=f(x)\frac{dA}{dx} = f(x)

Read that again: the area-so-far function is a function whose derivative is ff. That is exactly what an antiderivative is. So the machinery you built for reversing differentiation is, astonishingly, also the machinery for measuring area. That's why the same ∫\int symbol shows up in both jobs — it's not a coincidence, it's the Fundamental Theorem of Calculus.

The area between x=ax = a and x=bx = b is then "area collected by bb" minus "area collected by aa":

∫abf(x) dx=F(b)−F(a)\int_a^b f(x)\,dx = F(b) - F(a)

— top limit first, because you're subtracting the earlier total from the later one. Swap them and you get the negative of the right answer. And the +C+C cancels in the subtraction (every member of the antiderivative family gives the same difference), which is why definite integrals don't carry one.

What about a negative answer? When the curve dips below the xx-axis, the slivers have negative height, so the integral counts that region as negative. The integral is a signed total — "area above minus area below" — not raw area. If a question asks for actual area, find where the curve crosses the axis, integrate each piece separately, and add the sizes. A negative integral is not a mistake; ignoring the sign when the question says "area" is.1234-112345xyIn the shaded strip between x=1x = 1 and x=3x = 3 the curve dips below the axis, so the integral counts that region as negative — the signed area, not the raw area.