Limits of integration

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Limits of integration are the two numbers a and b that define the interval [a, b] over which you compute a definite integral ∫ from a to b of f(x) dx. They mark the start and end of the region (often the area under the curve) you’re measuring.

- Example: If f(x) = x^3 on the interval [2, 4], you compute ∫ from 2 to 4 of x^3 dx.

- Substitution: If you set u = g(x) with du = g′(x) dx, the limits change to u = g(a) and u = g(b). For instance, ∫ from 0 to 2 of 2x cos(x^2) dx becomes ∫ from 0 to 4 of cos(u) du when u = x^2.

- How the new limits are found: the lower bound becomes g(a) and the upper bound becomes g(b).

- Improper integrals (infinite limits): If the interval goes to infinity, you take a limit. For example, ∫ from a to ∞ f(x) dx = lim as t→∞ ∫ from a to t f(x) dx, and ∫ from −∞ to b f(x) dx = lim as t→−∞ ∫ from t to b f(x) dx.

- Splitting the interval: If c is between a and b, ∫ from a to b f(x) dx = ∫ from a to c f(x) dx + ∫ from c to b f(x) dx.