E078 Section 2 — Integration

Calculus 2 for Engineers (WGU E078), Section 2: approximating areas, definite integral, FTC, net change, u-substitution, exp/log and inverse-trig integrals. Built from the zyBooks Section 2 study materials; not official WGU material.

Sigma notation: what does $\sum_{i=1}^{n} a_i$ mean?

$a_1 + a_2 + \cdots + a_n$ (add up the terms as $i$ counts from $1$ to $n$)

$\sum_{i=1}^{n} c = \,?$

$n \cdot c$

$\sum_{i=1}^{n} i = \,?$

$\frac{n(n+1)}{2}$

$\sum_{i=1}^{n} i^2 = \,?$

$\frac{n(n+1)(2n+1)}{6}$

$\sum_{i=1}^{n} i^3 = \,?$

$\frac{n^2(n+1)^2}{4}$ (which equals $\left[\frac{n(n+1)}{2}\right]^2$)

$\Delta x$ for $n$ equal subintervals of $[a,b]$

$\Delta x = \frac{b - a}{n}$, and $x_i = a + i\,\Delta x$

Left-endpoint approximation $L_n$

$L_n = \sum_{i=1}^{n} f(x_{i-1})\,\Delta x$ (height from the left edge of each strip)

Right-endpoint approximation $R_n$

$R_n = \sum_{i=1}^{n} f(x_i)\,\Delta x$ (height from the right edge of each strip)

What is a Riemann sum?

$\sum f(x_i^*)\,\Delta x$ where $x_i^*$ is any point in the $i$-th subinterval

Upper sum vs. lower sum

Upper: use the max of $f$ on each strip. Lower: use the min. True area is between them.

If $f$ is increasing, which endpoint sum is the lower sum?

The left-endpoint sum $L_n$ (right-endpoint $R_n$ is the upper sum). Reversed if $f$ is decreasing.

Definition of the definite integral

$\int_a^b f(x)\,dx = \lim_{n\to\infty} \sum_{i=1}^{n} f(x_i^*)\,\Delta x$

What does a definite integral measure?

Net signed area: area above the $x$-axis minus area below it

How do you find TOTAL area between $f$ and the $x$-axis?

$\int_a^b |f(x)|\,dx$. Split at the zeros of $f$ and add the absolute values.

$\int_a^a f(x)\,dx = \,?$

$0$

$\int_b^a f(x)\,dx$ in terms of $\int_a^b f(x)\,dx$

$\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx$

Additivity property of integrals

$\int_a^b f = \int_a^c f + \int_c^b f$

Bounds property: if $m \le f(x) \le M$ on $[a,b]$, then...

$m(b - a) \le \int_a^b f(x)\,dx \le M(b - a)$

Average value of $f$ on $[a,b]$

$f_{\text{ave}} = \frac{1}{b - a} \int_a^b f(x)\,dx$

Mean Value Theorem for Integrals

If $f$ is continuous on $[a,b]$, there is a $c$ in $[a,b]$ with $f(c) = \frac{1}{b - a} \int_a^b f(x)\,dx$

FTC Part 1

If $F(x) = \int_a^x f(t)\,dt$, then $F'(x) = f(x)$

$\frac{d}{dx} \int_a^{g(x)} f(t)\,dt = \,?$

$f(g(x)) \cdot g'(x)$ (chain rule)

$\frac{d}{dx} \int_{h(x)}^{g(x)} f(t)\,dt = \,?$

$f(g(x))\,g'(x) - f(h(x))\,h'(x)$

$\frac{d}{dx} \int_x^5 f(t)\,dt = \,?$

$-f(x)$ (variable lower limit gives a minus sign)

FTC Part 2

$\int_a^b f(x)\,dx = F(b) - F(a)$, where $F$ is any antiderivative of $f$

Do you add $+C$ in a definite integral?

No. It cancels in $F(b) - F(a)$.

Power rule for integrals

$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$, $n \ne -1$

$\int \sin x\,dx = \,?$

$-\cos x + C$

$\int \cos x\,dx = \,?$

$\sin x + C$

$\int \sec^2 x\,dx = \,?$

$\tan x + C$

$\int \sec x \tan x\,dx = \,?$

$\sec x + C$

$\int \csc^2 x\,dx = \,?$

$-\cot x + C$

Net Change Theorem

$F(b) = F(a) + \int_a^b F'(x)\,dx$ (new value = old value + integral of the rate)

Integral of velocity over $[a,b]$ gives...

Displacement (net change in position)

How do you find total distance traveled from $v(t)$?

$\int_a^b |v(t)|\,dt$: find where $v = 0$, split there, add absolute values

Integral of an EVEN function over $[-a,a]$

$\int_{-a}^{a} f = 2\int_0^a f$

Integral of an ODD function over $[-a,a]$

$0$

Test for odd/even

Even: $f(-x) = f(x)$. Odd: $f(-x) = -f(x)$.

u-substitution is the reverse of which rule?

The chain rule

u-substitution formula

$\int f(g(x))\,g'(x)\,dx = \int f(u)\,du$ with $u = g(x)$, $du = g'(x)\,dx$

Definite integral with u-sub: what happens to the limits?

Change them: $x = a \to u = g(a)$, $x = b \to u = g(b)$. Don't switch back to $x$.

$\int x\sqrt{x-1}\,dx$: what's the trick?

Let $u = x - 1$ AND rewrite $x = u + 1$, then integrate $(u+1)u^{1/2}$

$\int e^x\,dx = \,?$

$e^x + C$

$\int e^{kx}\,dx = \,?$

$\frac{1}{k}e^{kx} + C$

$\int a^x\,dx = \,?$

$\frac{a^x}{\ln a} + C$

$\int \frac{1}{x}\,dx = \,?$

$\ln|x| + C$

$\int \frac{g'(x)}{g(x)}\,dx = \,?$

$\ln|g(x)| + C$

$\int \ln x\,dx = \,?$

$x \ln x - x + C$

$\int \tan x\,dx = \,?$

$\ln|\sec x| + C$ (which equals $-\ln|\cos x| + C$)

$\int \frac{du}{\sqrt{a^2 - u^2}} = \,?$

$\sin^{-1}\!\left(\frac{u}{a}\right) + C$

$\int \frac{du}{a^2 + u^2} = \,?$

$\frac{1}{a} \tan^{-1}\!\left(\frac{u}{a}\right) + C$

$\int \frac{du}{u\sqrt{u^2 - a^2}} = \,?$

$\frac{1}{a} \sec^{-1}\!\left(\frac{|u|}{a}\right) + C$

Which inverse-trig formula has NO $\frac{1}{a}$ in front?

Arcsine: $\int \frac{du}{\sqrt{a^2 - u^2}} = \sin^{-1}\!\left(\frac{u}{a}\right) + C$

$\int_0^1 \frac{x}{1+x^4}\,dx$: what substitution?

$u = x^2$, $du = 2x\,dx \to \frac{1}{2}\int_0^1 \frac{du}{1+u^2} = \frac{\pi}{8}$