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Integration

Area is a sum of very thin slices.

How much space sits under a curve? There's no formula for a curved shape — so cheat. Fill it with rectangles, add them up, and then make them thinner.

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Four rectangles, each 0.50.5 wide, each as tall as the curve at its middle. Together they come to 4.625 — close, but look at the tops: every one pokes above the curve on one side and falls short on the other.

Key result

∫abf(x) dx=lim⁡δx→0∑f(x) δx\int_a^b f(x)\,dx = \lim_{\delta x \to 0}\sum f(x)\,\delta x

Thinner rectangles, smaller errors: 4.6563, 4.6641, 4.6665. The totals home in on one number — that limit is the definite integral. The ∫\textstyle\int is a stretched S, for sum.

The rule

∫xn dx=xn+1n+1+c(n≠−1)\begin{gathered} \int x^n\,dx = \frac{x^{n+1}}{n+1} + c \\ (n \neq -1) \end{gathered}

The shortcut runs differentiation backwards: raise the power by one, then divide by the new power. Check it — differentiate x33+x\tfrac{x^3}{3} + x and x2+1x^2 + 1 comes straight back.

Chapter 14

Every rectangle, added up at once.

The constant of integration, finding a curve from its gradient, area between a curve and a line, and twelve graded exam questions with full worked solutions are waiting.

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