Notes
Accumulation and Riemann sums
The area under a rate gives the accumulated change. If $r(t)$ is liters per minute, $\displaystyle \int_a^b r(t)\,dt$ is liters gained from $a$ to $b$.
- Left, right and midpoint Riemann sums use rectangles; the trapezoidal sum uses trapezoids. Intervals can be unequal in a table.
- For increasing $f$: left sums underestimate and right sums overestimate. For concave up $f$: the trapezoidal sum overestimates.
- The definite integral is the limit of Riemann sums: $\displaystyle \int_a^b f(x)\,dx = \lim_{n \to \infty}\sum_{k=1}^{n} f(x_k)\Delta x$.
Properties
- $\displaystyle \int_a^a f = 0$, $\displaystyle \int_b^a f = -\int_a^b f$, $\displaystyle \int_a^b f + \int_b^c f = \int_a^c f$.
- Area below the axis counts as negative.
The Fundamental Theorem of Calculus
- Part 1: $\displaystyle \frac{d}{dx}\int_a^x f(t)\,dt = f(x)$. With a chain: $\displaystyle \frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x))g'(x)$.
- Part 2: $\displaystyle \int_a^b f(x)\,dx = F(b) - F(a)$, where $F' = f$.
- Accumulation: $\displaystyle f(b) = f(a) + \int_a^b f'(x)\,dx$.
Antiderivatives
| $f(x)$ | $\int f(x)\,dx$ | ||
|---|---|---|---|
| $x^n$, $n \ne -1$ | $\displaystyle \frac{x^{n+1}}{n+1} + C$ | ||
| $\displaystyle \frac{1}{x}$ | $\ln | x | + C$ |
| $e^x$ | $e^x + C$ | ||
| $\sin x$ | $-\cos x + C$ | ||
| $\cos x$ | $\sin x + C$ | ||
| $\sec^2 x$ | $\tan x + C$ | ||
| $\displaystyle \frac{1}{1 + x^2}$ | $\tan^{-1}x + C$ |
u-substitution
Pick $u$ as an "inside" function whose derivative also appears. $\displaystyle \int 2x\cos(x^2)\,dx$: $u = x^2$, $du = 2x\,dx$, giving $\sin(x^2) + C$. For definite integrals, change the bounds to $u$-values.
BC only: more integration techniques
- Integration by parts: $\displaystyle \int u\,dv = uv - \int v\,du$. Choose $u$ to get simpler when differentiated (logs, then polynomials, then trig and exponentials). Example: $\displaystyle \int xe^x\,dx = xe^x - e^x + C$.
- Partial fractions (non-repeating linear factors): $\displaystyle \frac{1}{(x-1)(x+2)} = \frac{1/3}{x - 1} - \frac{1/3}{x + 2}$, then integrate each to a log.
- Improper integrals: an infinite bound or an infinite discontinuity. Replace it with a limit: $\displaystyle \int_1^\infty \frac{1}{x^2}\,dx = \lim_{b \to \infty}\left(1 - \frac{1}{b}\right) = 1$, so it converges. $\displaystyle \int_1^\infty \frac{1}{x}\,dx$ diverges.
Topics in this unit
Each topic has its own short notes and practice.
- 6.1Exploring Accumulations of Change
- 6.2Approximating Areas with Riemann Sums
- 6.3Riemann Sums, Summation Notation, and Definite Integral Notation
- 6.4The Fundamental Theorem of Calculus and Accumulation Functions
- 6.5Interpreting the Behavior of Accumulation Functions Involving Area
- 6.6Applying Properties of Definite Integrals
- 6.7The Fundamental Theorem of Calculus and Definite Integrals
- 6.8Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation
- 6.9Integrating Using Substitution
- 6.10Integrating Functions Using Long Division and Completing the Square
- 6.11Integrating Using Integration by Parts
- 6.12Integrating Using Linear Partial Fractions
- 6.13Evaluating Improper Integrals
- 6.14Selecting Techniques for Antidifferentiation
Practice
$[x^2 + x]_1^3 = 12 - 2 = 10$.
FTC Part 1: the derivative of the accumulation function is the integrand at $x$.
Chain rule with FTC: $\cos(x^2) \cdot 2x$.
$u = x^3 + 1$, $du = 3x^2\,dx$: $\displaystyle \int u^4\,du = \frac{u^5}{5}$.
For an increasing function, each left rectangle sits below the curve.
$\displaystyle f(4) = f(1) + \int_1^4 f'(x)\,dx = 5 + 7$.
The power rule fails for $n = -1$; the antiderivative of $\displaystyle \frac{1}{x}$ is $\ln|x|$.
The integral of a rate is the total change over the interval. The tank may have had water at $t = 0$.
Parts with $u = x$, $dv = \cos x\,dx$: $\displaystyle x\sin x - \int\sin x\,dx = x\sin x + \cos x + C$.
$\displaystyle \lim_{b \to \infty}\left[-\frac{1}{2x^2}\right]_1^b = 0 + \frac{1}{2}$.
$\displaystyle \frac{1}{(x-1)(x+1)} = \frac{1/2}{x - 1} - \frac{1/2}{x + 1}$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
The rate that people enter a museum is given by a table: $r(0) = 0$, $r(2) = 30$, $r(4) = 50$, $r(7) = 60$, with $t$ in hours and $r$ in people per hour.
(a) Use a trapezoidal sum with the three subintervals to approximate $\int_0^7 r(t)\,dt$.
Show the answer and scoring guide
$\displaystyle 2 \cdot \frac{0 + 30}{2} + 2 \cdot \frac{30 + 50}{2} + 3 \cdot \frac{50 + 60}{2} = 30 + 80 + 165 = 275$.
Check off the points you earned:
(b) Explain the meaning of $\int_0^7 r(t)\,dt$ in context.
Show the answer and scoring guide
It's the total number of people who entered the museum from $t = 0$ to $t = 7$ hours.
Check off the points you earned:
(c) Another model is $R(t) = 60 - 60e^{-0.3t}$. Write, but don't evaluate, an expression for the average rate of entry over $0 \le t \le 7$.
Show the answer and scoring guide
$\displaystyle \frac{1}{7}\int_0^7 \left(60 - 60e^{-0.3t}\right)dt$ people per hour.
Check off the points you earned: