AP Calculus AB and BC glossary

Changing limits of integration

Also called: Changing the bounds

Changing the limits of integration means converting the x values on a definite integral into the matching u values when you substitute. You then evaluate the new antiderivative at the new limits, so there is no need to rewrite the answer in terms of x.

Run the substitution through the bounds as well as the integrand. With $u = x^2$ and $du = 2x\,dx$ , the bound $x = 0$ becomes $u = 0$ and the bound $x = 2$ becomes $u = 4$ , so the whole problem moves into $u$ and stays there.

\int_0^2 x e^{x^2}\,dx = \frac{1}{2}\int_0^4 e^u\,du = \frac{1}{2}\left(e^4 - 1\right)

Occasionally converting finishes the problem outright. In $\int_{-1}^{1} x e^{x^2}\,dx$ the same substitution sends both bounds to $u = 1$ , and an integral whose two limits agree is zero, so no antiderivative is needed.

The mistake

Doing both halves at once. Students convert the bounds to $u$ values, then also rewrite the antiderivative back in terms of $x$ , and evaluate $\frac{1}{2}e^{x^2}$ at 0 and 4. Pick one lane: evaluate in $u$ at the $u$ bounds, or return to $x$ and use the original $x$ bounds.

Appears in: Unit 6: Integration and Accumulation