Integration gives a precise language for measuring accumulation, with practical applications in geometry. The area between two curves \(y = f(x)\) and \(y = g(x)\) is given by \(\int_a^b |f(x) - g(x)| \, dx\), where \(a\) and \(b\) are the points of intersection. When \(f(x) \geq g(x)\) throughout, this simplifies to \(\int_a^b [f(x) - g(x)] \, dx\). The first step in such problems is finding the intersection points by setting \(f(x) = g(x)\) and solving. For example, the area between \(y = x^2\) and \(y = x\) on \([0, 1]\) is \(\int_0^1 (x - x^2) \, dx = \frac{1}{6}\). If the curves cross within the interval, the integral must be split at each crossing point so that positive and negative contributions do not cancel. When the curves are easier to describe as functions of \(y\), one can integrate with respect to \(y\) instead, using \(\int_c^d |f(y) - g(y)| \, dy\) with horizontal slices.
When a region is revolved around an axis, the resulting three-dimensional solid has a volume expressible as an integral. The disk method applies when cross sections perpendicular to the axis of rotation are solid disks: \(V = \pi \int_a^b [R(x)]^2 \, dx\), where \(R(x)\) is the distance from the curve to the axis. The washer method handles solids with a hole, giving \(V = \pi \int_a^b ([R(x)]^2 - [r(x)]^2) \, dx\) with outer radius \(R\) and inner radius \(r\). When the axis of rotation is perpendicular to the variable of integration, the shell method is often easier: \(V = 2\pi \int_a^b x \cdot f(x) \, dx\) for revolution about the \(y\)-axis, where each cylindrical shell has radius \(x\), height \(f(x)\), and thickness \(dx\). For example, revolving \(y = \sqrt{x}\) from \(x = 0\) to \(x = 4\) about the \(x\)-axis gives \(V = \pi \int_0^4 x \, dx = 8\pi\). Revolving about a non-standard axis simply adjusts the radius function to measure distance from the new axis.
Indeterminate forms arise when limits cannot be evaluated by direct substitution. L'Hopital's Rule addresses the cases \(0/0\) and \(\infty/\infty\): if \(\lim_{x \to c} f(x)/g(x)\) is indeterminate of this kind, then the limit equals \(\lim_{x \to c} f'(x)/g'(x)\), provided this latter limit exists. The rule can be applied repeatedly if necessary. Other indeterminate forms \((0 \cdot \infty\), \(\infty - \infty\), \(0^0\), \(1^\infty\), \(\infty^0)\) must first be rewritten into \(0/0\) or \(\infty/\infty\) form through algebraic manipulation. For instance, \(\lim_{x \to 0^+} x \ln x\) is rewritten as \(\lim_{x \to 0^+} \frac{\ln x}{1/x}\), after which L'Hopital's Rule applies to give 0. The rule also confirms that \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) and shows that exponential growth dominates any polynomial: \(\lim_{x \to \infty} \frac{x}{e^x} = 0\).