MATH 101 Flashcards: Integrals, Series and Key Formulas

Question 1

How do you differentiate an integral with variable limits?

Accepted Answer

$\dfrac{d}{dx}\displaystyle\int_{g(x)}^{h(x)} f(t)\,dt = f(h(x))\,h'(x) - f(g(x))\,g'(x)$
This is FTC Part 1 with the chain rule. A constant limit adds nothing.

Question 2

What is the integral of an odd function over a symmetric interval?

Accepted Answer

Zero: $\displaystyle\int_{-a}^{a} f(x)\,dx = 0$ when $f(-x) = -f(x)$.
For an even function, $f(-x) = f(x)$, double the right half: $\displaystyle\int_{-a}^{a} f = 2\int_0^a f$.

Question 3

What is the integration by parts formula?

Accepted Answer

$\displaystyle\int u\,dv = uv - \int v\,du$
It is the product rule run backwards. You differentiate $u$ and integrate $dv$.

Question 4

What is the integral of $\sec x$?

Accepted Answer

$\displaystyle\int\sec x\,dx = \log|\sec x + \tan x| + C$
It often shows up after a $\tan$ or $\sec$ substitution. Some questions give it to you, but not all.

Question 5

What is the integral of $\tan x$?

Accepted Answer

$\displaystyle\int\tan x\,dx = \log|\sec x| + C$
This equals $-\log|\cos x| + C$. To see it, write $\tan x = \dfrac{\sin x}{\cos x}$ and let $u = \cos x$.

Question 6

How do you integrate $\sin^2 x$ or $\cos^2 x$?

Accepted Answer

Use $\sin^2 x = \tfrac12(1 - \cos 2x)$ and $\cos^2 x = \tfrac12(1 + \cos 2x)$, then integrate term by term.

Question 7

Which trig substitution do you use for each square root?

Accepted Answer

$\sqrt{a^2 - x^2}$: $x = a\sin\theta$. $\ \sqrt{a^2 + x^2}$: $x = a\tan\theta$. $\ \sqrt{x^2 - a^2}$: $x = a\sec\theta$.
Each one turns the inside into a perfect square: $a^2\cos^2\theta$, $a^2\sec^2\theta$, $a^2\tan^2\theta$.

Question 8

What partial fraction terms does each factor of the denominator get?

Accepted Answer

$x - a$: $\dfrac{A}{x - a}$. $\ (x - a)^k$: $\dfrac{A_1}{x - a} + \dots + \dfrac{A_k}{(x - a)^k}$. Irreducible quadratic: $\dfrac{Bx + D}{x^2 + bx + c}$.
This works only when the top's degree is less than the bottom's. If not, divide first.

Question 9

What is $\displaystyle\int \dfrac{dx}{x^2 + a^2}$?

Accepted Answer

$\dfrac{1}{a}\arctan\dfrac{x}{a} + C$
You need it for the irreducible-quadratic pieces in partial fractions. If there is an $x$ term, complete the square first.

Question 10

What is the formula for a right Riemann sum?

Accepted Answer

$\displaystyle\sum_{i=1}^{n} f(a + i\,\Delta x)\,\Delta x$, with $\Delta x = \dfrac{b-a}{n}$.
Left endpoints use $a + (i - 1)\Delta x$. Midpoints use $a + \left(i - \tfrac{1}{2}\right)\Delta x$. As $n \to \infty$, the sum goes to $\int_a^b f(x)\,dx$.

Question 11

What is Simpson's rule?

Accepted Answer

$S_n = \dfrac{\Delta x}{3}\big[f(x_0) + 4f(x_1) + 2f(x_2) + \dots + 4f(x_{n-1}) + f(x_n)\big]$, with $n$ even.
It is exact for polynomials of degree at most $3$.

Question 12

What are the error bounds for the midpoint, trapezoidal and Simpson's rules?

Accepted Answer

$|E_M| \le \dfrac{M(b-a)^3}{24n^2}$, $\ |E_T| \le \dfrac{M(b-a)^3}{12n^2}$, $\ |E_S| \le \dfrac{L(b-a)^5}{180n^4}$
Here $M \ge |f''(x)|$ and $L \ge |f^{(4)}(x)|$ for every $x$ in $[a, b]$. The midpoint bound is half the trapezoidal one.

Question 13

When does $\displaystyle\int_1^\infty \frac{dx}{x^p}$ converge?

Accepted Answer

It converges exactly when $p > 1$.
At $p = 1$ it diverges: $\int_1^R \frac{dx}{x} = \log R \to \infty$. For $p > 1$ its value is $\frac{1}{p - 1}$.

Question 14

When does $\displaystyle\int_0^1 \frac{dx}{x^p}$ converge?

Accepted Answer

Exactly when $p < 1$.
This is the opposite of the test at infinity. So $\int_0^\infty \frac{dx}{x^p}$ diverges for every $p$.

Question 15

What does the comparison test for improper integrals say?

Accepted Answer

If $|f(x)| \le g(x)$ and $\int g$ converges, then $\int f$ converges. If $f(x) \ge g(x) \ge 0$ and $\int g$ diverges, then $\int f$ diverges.
Smaller than a convergent integral means convergent. Bigger than a divergent one means divergent. The other two cases tell you nothing.

Question 16

What is the formula for the area between two curves?

Accepted Answer

$\displaystyle\int_a^b (\text{top} - \text{bottom})\,dx$, or $\displaystyle\int_c^d (\text{right} - \text{left})\,dy$.
The limits are where the curves meet. If they cross in between, split there.

Question 17

What is the formula for the washer method?

Accepted Answer

$V = \pi\displaystyle\int_a^b \big(R^2 - r^2\big)\,dx$
$R$ and $r$ are distances from the axis of rotation to the outer and inner curves. With no hole, $r = 0$ and it is the disk method, $\pi\int R^2\,dx$.

Question 18

How do you find the average value of a function on an interval?

Accepted Answer

$\bar f = \dfrac{1}{b - a}\displaystyle\int_a^b f(x)\,dx$
Integrate, then divide by the length of the interval.

Question 19

What is Hooke's law for a spring?

Accepted Answer

$F = kx$, where $x$ is the stretch beyond natural length.
So the work to stretch from $x = a$ to $x = b$ is $\displaystyle\int_a^b kx\,dx = \tfrac{k}{2}\left(b^2 - a^2\right)$. Measure $x$ in meters.

Question 20

How do you find the center of mass of a rod with density $\rho(x)$?

Accepted Answer

$\bar x = \dfrac{\int_a^b x\,\rho(x)\,dx}{\int_a^b \rho(x)\,dx}$
It is moment divided by mass, as for point masses, with integrals instead of sums.

Question 21

What are the formulas for the centroid of a region between two curves?

Accepted Answer

$\bar x = \dfrac{1}{A}\displaystyle\int_a^b x\,(T - B)\,dx$ and $\bar y = \dfrac{1}{2A}\displaystyle\int_a^b \left(T^2 - B^2\right)dx$
$T$ is the top curve, $B$ the bottom one and $A$ the area. In $\bar y$ it is $T^2 - B^2$, not $(T - B)^2$.

Question 22

How do you solve a separable differential equation?

Accepted Answer

Write $\dfrac{dy}{dx} = f(x)\,g(y)$, then $\displaystyle\int \dfrac{dy}{g(y)} = \int f(x)\,dx + C$.
One constant is enough. Use the initial condition only after integrating. If $g(c) = 0$, then $y = c$ is also a solution.

Question 23

Which grows fastest as $n \to \infty$: $\log n$, $n^p$, $a^n$ or $n!$?

Accepted Answer

$\log n \ll n^p \ll a^n \ll n!$ (for $p > 0$, $a > 1$)
In a fraction, divide top and bottom by the fastest term in the bottom. Each slower term then goes to 0.

Question 24

What is the limit of $\left(1 + \dfrac{a}{n}\right)^n$ as $n \to \infty$?

Accepted Answer

$e^a$
This is a $1^\infty$ form, so don't guess $1$. Take $\log$, use l'Hôpital's rule, then raise $e$ to that power.

Question 25

What is the sum of a geometric series?

Accepted Answer

$\displaystyle\sum_{n=0}^{\infty} ar^{n} = \dfrac{a}{1 - r}$ when $|r| < 1$. When $|r| \geq 1$ it diverges.
In general: first term over $1 - r$. A series that starts at $n = 2$ has first term $ar^{2}$.

Question 26

When does a p-series converge?

Accepted Answer

$\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^{p}}$ converges if $p > 1$ and diverges if $p \leq 1$.
So the harmonic series $\sum \dfrac{1}{n}$ ($p = 1$) diverges. These are the usual series to compare against.

Question 27

What does the divergence test say?

Accepted Answer

If $\lim_{n\to\infty} a_n \neq 0$ (or the limit does not exist), $\sum a_n$ diverges. If the limit is $0$, the test says nothing.
It can only prove divergence: $\dfrac{1}{n} \to 0$, yet $\sum \dfrac{1}{n}$ diverges.

Question 28

What does the ratio test say?

Accepted Answer

$L = \displaystyle\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|$: $L < 1$ converges, $L > 1$ diverges, $L = 1$ tells you nothing.
It works well for factorials and powers like $5^{n}$ or $n^{n}$. Rational terms and $p$-series always give $L = 1$, so compare those instead.

Question 29

How big is the error when you stop an alternating series early?

Accepted Answer

At most the first term left out: $|S - S_N| \leq b_{N+1}$.
It holds when the signs alternate and $b_n$ decreases to $0$, the conditions of the alternating series test.

Question 30

What is the formula for a Taylor series?

Accepted Answer

$\displaystyle\sum_{n=0}^{\infty} \dfrac{f^{(n)}(a)}{n!}(x - a)^{n}$. A Maclaurin series is the case $a = 0$.
Read it backward to get derivatives: $f^{(n)}(a) = n!$ times the coefficient of $(x - a)^{n}$.

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