Integration and Accumulation of Change

Where a derivative measures a rate, an integral 积分 measures an accumulation 累积 – a total built up from a rate. If a rate of change acts over an interval, the area between its graph and the axis gives the net accumulated change. This "area = total change" idea is the foundation of integral calculus.

A Riemann sum 黎曼和 estimates the area under a curve by adding the areas of thin rectangles. Split $[a,b]$ into subintervals and use the function's height at the left endpoint, right endpoint, or midpoint of each. A trapezoidal sum 梯形法 uses trapezoids instead, averaging the two endpoint heights – usually more accurate. With more, thinner rectangles the estimate improves.

A Riemann sum approximates the area under a curve with rectangles

Strips of width h approximate the area under a curve

Exam skill: be able to compute left, right, midpoint, and trapezoidal estimates from a table or graph, and state whether each over- or under-estimates based on whether the function is increasing/decreasing or concave up/down.

Writing a Riemann sum with summation notation $\sum_{k=1}^{n} f(x_k)\,\Delta x$ and letting the number of rectangles grow without bound gives the exact area – the definite integral 定积分:

The integral is the limit of Riemann sums; $a$ and $b$ are the limits of integration.

An accumulation function 累积函数 $g(x)=\int_a^x f(t)\,dt$ gives the accumulated area from $a$ up to $x$. The Fundamental Theorem of Calculus (FTC) 微积分基本定理 says its derivative is the integrand:

Differentiation and integration are inverse operations. With a variable upper limit and the chain rule, $\dfrac{d}{dx}\int_a^{u(x)} f(t)\,dt=f(u(x))\,u'(x)$.

The accumulation function adds signed area; the FTC says its derivative is f

Because $g'(x)=f(x)$, the graph of $f$ tells you everything about $g$: $g$ increases where $f>0$, decreases where $f<0$, has extrema where $f$ crosses zero, and is concave up where $f$ is increasing. Reading these connections off a graph of $f$ is a classic free-response task.

Definite integrals obey useful rules: reversing the limits negates the value ($\int_b^a=-\int_a^b$), an integral over a zero-width interval is $0$, they add over adjacent intervals ($\int_a^c=\int_a^b+\int_b^c$), and constants factor out. Use these to combine or split given integral values.

The evaluation form of the FTC computes a definite integral from an antiderivative 原函数 $F$ (where $F'=f$):

So integrating a rate of change over $[a,b]$ gives the net change in the quantity – the single most-used result in the course.

Worked example. $\displaystyle\int_1^3 (2x+1)\,dx$: an antiderivative is $F(x)=x^2+x$, so the value is $F(3)-F(1)=12-2=10$.

A definite integral is the signed area between the curve and the x-axis

An indefinite integral 不定积分 $\int f(x)\,dx=F(x)+C$ is the family of all antiderivatives (hence the constant of integration $C$). Reverse each derivative rule: the power rule becomes $\int x^n\,dx=\dfrac{x^{n+1}}{n+1}+C$ (for $n\neq-1$), with $\int \frac1x\,dx=\ln|x|+C$, and the antiderivatives of $e^x$, $\sin x$, $\cos x$, and $\sec^2 x$ come straight from their derivatives.

u-substitution 换元积分 reverses the chain rule: choose $u=g(x)$ so that $g'(x)$ also appears, turning $\int f(g(x))g'(x)\,dx$ into $\int f(u)\,du$. Remember to convert $dx$ to $du$ and, for a definite integral, either change the limits to $u$-values or convert back to $x$ at the end.

When a rational integrand is "top-heavy" (numerator degree $\ge$ denominator degree), long division rewrites it as a polynomial plus a proper fraction you can integrate. Completing the square in a denominator turns it into a form like $u^2+a^2$, leading to an arctangent antiderivative $\frac1a\arctan\frac{u}{a}+C$.

Integration by parts 分部积分 reverses the product rule:

Choose $u$ to simplify when differentiated and $dv$ to be easy to integrate (the LIATE guide: logs, inverse-trig, algebraic, trig, exponential). It handles products like $\int x e^x\,dx$ and $\int x\ln x\,dx$, sometimes applied twice.

Worked example. For $\int x e^x\,dx$, choose $u=x$ ($du=dx$) and $dv=e^x\,dx$ ($v=e^x$):

Partial fractions 部分分式 split a rational function with a factorable denominator into a sum of simpler fractions:

each of which integrates to a logarithm. This technique is essential for the logistic differential equation in the next unit.

An improper integral 反常积分 has an infinite limit of integration or an infinite discontinuity in the integrand. Evaluate it as a limit: $\int_a^\infty f\,dx=\lim_{b\to\infty}\int_a^b f\,dx$. If the limit is a finite number the integral converges 收敛; otherwise it diverges 发散.

Worked example. $\displaystyle\int_1^\infty \frac{1}{x^2}\,dx=\lim_{b\to\infty}\left[-\frac1x\right]_1^b=\lim_{b\to\infty}\left(1-\frac1b\right)=1$, so it converges to $1$. By contrast $\int_1^\infty \frac1x\,dx$ gives $\lim_{b\to\infty}\ln b=\infty$ and diverges – the same integrand-shape can go either way.

The BC exam expects you to recognize which method fits: basic rules, substitution (a chain-rule pattern), by parts (a product), partial fractions (a factorable rational), or long division/completing the square. Being able to look at an integral and pick the right tool quickly is itself a tested skill.

• Integration is antidifferentiation; use the power rule $\int x^n\,dx=\tfrac{x^{n+1}}{n+1}+C$ and don't forget the $+C$.
• The Fundamental Theorem links the two: $\int_a^b f'(x)\,dx=f(b)-f(a)$, and $\tfrac{d}{dx}\int_a^x f(t)\,dt=f(x)$.
• Approximate a definite integral with Riemann sums or the trapezoidal rule from a table of values.
• A definite integral is a signed area (below the axis counts negative); split at sign changes for total area.
• Use u-substitution and remember to change the limits (or back-substitute) accordingly.