E078 Section 2 Notes – Integration

Source: Condensed from the Section 2 Study Guide (zyBooks 2.1–2.7). Not official WGU material.

Big idea. Integration starts as a limit of Riemann sums and becomes the exact "net signed area" under a curve. The Fundamental Theorem of Calculus turns that limit into an antiderivative computation; the rest of the section is a toolkit for antiderivatives: basic formulas, substitution, exponential/log patterns, and inverse-trig patterns.

2.1 Approximating Areas

Big idea

• Sigma notation $\sum$ means "add these up."

• Estimate area under a curve: chop $[a,b]$ into $n$ equal strips of width $\Delta x$ and add rectangle areas.

• $L_n$: height is $f$ at the left edge of each strip. $R_n$: height is $f$ at the right edge. Any point $x_i^*$ in each strip gives a Riemann sum; bigger $n$ is better.

• Upper sum: use the max of $f$ on each strip. Lower sum: use the min.

Formulas

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

• $L_n = \sum_{i=1}^{n} f(x_{i-1})\,\Delta x$

• $R_n = \sum_{i=1}^{n} f(x_i)\,\Delta x$

• Riemann sum: $\sum_{i=1}^{n} f(x_i^*)\,\Delta x$

• $\sum_{i=1}^{n} c = nc$, $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$, $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$, $\sum_{i=1}^{n} i^3 = \frac{n^2(n+1)^2}{4}$

• $\sum (a_i \pm b_i) = \sum a_i \pm \sum b_i$, $\sum c\,a_i = c \sum a_i$

Worked example

• Sigma sum: $\sum_{i=1}^{10} (3i^2 + 2) = 3(385) + 20 = 1175$, using $\sum_{i=1}^{10} i^2 = \frac{10\cdot 11\cdot 21}{6} = 385$.

• Endpoints: $f(x) = x^3$ on $[0,2]$, $n = 4$, $\Delta x = 0.5$; values $0, 0.125, 1, 3.375, 8$. $L_4 = 0.5(0 + 0.125 + 1 + 3.375) = 2.25$; $R_4 = 0.5(0.125 + 1 + 3.375 + 8) = 6.25$. Increasing $f$, so $L_4$ is lower, $R_4$ is upper; true area $4$ is between.

Common mistakes

• Using all $n+1$ points: $L_n$ skips the last, $R_n$ skips the first.

• Forgetting to multiply by $\Delta x$; writing $\sum_{i=1}^{n} c = c$ instead of $nc$.

• Assuming left means lower. True only for increasing $f$; for decreasing $f$, left is upper.

2.2 The Definite Integral

Big idea

• The definite integral is the limit of Riemann sums: exact "net signed area" (above the $x$-axis counts positive, below counts negative).

• Total area uses the absolute value: $\int_a^b |f(x)|\,dx$.

• Some integrals can be done with geometry (triangles, rectangles, circles).

• Average value equals the integral divided by the interval length.

Formulas

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

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

• Properties: $\int_a^a f = 0$; $\int_b^a f = -\int_a^b f$; $\int_a^b [f \pm g] = \int_a^b f \pm \int_a^b g$; $\int_a^b c\,f = c\int_a^b f$; $\int_a^b f = \int_a^c f + \int_c^b f$

• Comparison: if $f \ge g$ on $[a,b]$ then $\int_a^b f \ge \int_a^b g$; if $m \le f \le M$ then $m(b-a) \le \int_a^b f \le M(b-a)$

Worked example

• Geometry: $\int_{-2}^{2} \sqrt{4 - x^2}\,dx$ is the top half of a circle of radius $2$: area $\frac{1}{2}\pi(2)^2 = 2\pi$.

• Net vs. total: $f(x) = x - 1$ on $[0,4]$, zero at $x = 1$. Triangle below on $[0,1]$: $0.5$; above on $[1,4]$: $4.5$. Net $= 4$; total $= 5$.

• Average value: $f(x) = x^2$ on $[0,3]$: $\int_0^3 x^2\,dx = 9$, so $f_{\text{ave}} = \frac{9}{3-0} = 3$.

Common mistakes

• Mixing up net and total area: for "total area," split at the zeros and use absolute values.

• Forgetting to divide by $(b-a)$ for average value; flipping limits without flipping the sign.

2.3 The Fundamental Theorem of Calculus (FTC)

Big idea

• MVT for Integrals: a continuous $f$ actually hits its average value at some $c$ in $[a,b]$.

• FTC Part 1: the derivative of an "area so far" function is the original function. Plug in the upper limit and multiply by its derivative (chain rule).

• FTC Part 2: to evaluate $\int_a^b f$, find any antiderivative $F$ and compute $F(b) - F(a)$.

Formulas

• MVT: $f(c) = \frac{1}{b-a}\int_a^b f(x)\,dx$ for some $c \in [a,b]$

• FTC 1: $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$; with chain rule: $\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x))\,g'(x)$

• Two variable limits: $\frac{d}{dx}\int_{h(x)}^{g(x)} f(t)\,dt = f(g(x))\,g'(x) - f(h(x))\,h'(x)$

• FTC 2: $\int_a^b f(x)\,dx = F(b) - F(a)$ where $F' = f$

Worked example

• MVT: $f(x) = x^2$ has average value $3$ on $[0,3]$ (from 2.2). Solve $c^2 = 3$: $c = \sqrt{3} \approx 1.73$, in $[0,3]$.

• FTC 1 with chain rule: $\frac{d}{dx}\int_0^{\sqrt{x}} \sin t\,dt = \sin(\sqrt{x})\cdot\frac{1}{2\sqrt{x}}$.

• Two variable limits: $\frac{d}{dx}\int_x^{2x} t^3\,dt = (2x)^3\cdot 2 - x^3\cdot 1 = 15x^3$.

• FTC 2: $\int_0^{\pi} \sin x\,dx = \left[-\cos x\right]_0^{\pi} = 1 + 1 = 2$.

Common mistakes

• Forgetting the chain-rule factor $g'(x)$ when the limit is not just $x$.

• A variable lower limit gives a minus sign: $\frac{d}{dx}\int_x^5 f(t)\,dt = -f(x)$.

• Writing $F(a) - F(b)$ in FTC 2; it is always top minus bottom. Adding $+C$ to a definite integral (it cancels).

2.4 Integration Formulas and the Net Change Theorem

Big idea

• Integrating a rate of change gives the net change: new value = old value + integral of the rate.

• Integrating velocity gives displacement (net); integrating $|v(t)|$ gives total distance.

• Even ($f(-x) = f(x)$): double the right half. Odd ($f(-x) = -f(x)$): cancels to $0$ on $[-a,a]$.

Formulas

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

• $\int \cos x\,dx = \sin x + C$, $\int \sin x\,dx = -\cos x + C$, $\int \sec^2 x\,dx = \tan x + C$, $\int \sec x \tan x\,dx = \sec x + C$, $\int \csc^2 x\,dx = -\cot x + C$

• Net change: $F(b) = F(a) + \int_a^b F'(x)\,dx$

• Even: $\int_{-a}^{a} f = 2\int_0^a f$. Odd: $\int_{-a}^{a} f = 0$

Worked example

• $v(t) = 3t - 5$ m/s on $[0,3]$. Displacement: $\int_0^3 (3t - 5)\,dt = \left[\frac{3}{2}t^2 - 5t\right]_0^3 = 13.5 - 15 = -1.5$ m.

• Total distance: $v = 0$ at $t = \frac{5}{3}$. $\int_0^{5/3} (3t-5)\,dt = -\frac{25}{6}$, $\int_{5/3}^{3} (3t-5)\,dt = \frac{8}{3}$, so total $= \frac{25}{6} + \frac{16}{6} = \frac{41}{6} \approx 6.83$ m.

• Even/odd: $\int_{-2}^{2} 3x^4\,dx = 2\int_0^2 3x^4\,dx = \frac{192}{5}$; $\int_{-\pi}^{\pi} x^2 \sin x\,dx = 0$ since $(-x)^2\sin(-x) = -x^2\sin x$ (odd).

Common mistakes

• Answering "total distance" with the displacement; find where $v = 0$ first.

• Calling a function odd without checking $f(-x) = -f(x)$ (a constant term makes it not odd). The odd shortcut only works on a symmetric interval $[-a,a]$.

2.5 Substitution (u-sub)

Big idea

• Substitution is the chain rule in reverse: find an inside function $u = g(x)$ whose derivative $g'(x)$ is also in the integrand (up to a constant).

• Replace everything with $u$ and $du$; no $x$ should be left.

• For a definite integral, change the limits to $u$-values and never go back to $x$.

Formulas

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

• $\int_a^b f(g(x))\,g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du$

Worked example

• Indefinite: $\int 6x(3x^2+4)^4\,dx$, $u = 3x^2 + 4$, $du = 6x\,dx$: $\int u^4\,du = \frac{(3x^2+4)^5}{5} + C$.

• Leftover $x$ trick: $\int x\sqrt{x-1}\,dx$, $u = x - 1$, $x = u + 1$: $\int (u^{3/2} + u^{1/2})\,du = \frac{2}{5}(x-1)^{5/2} + \frac{2}{3}(x-1)^{3/2} + C$.

• Definite: $\int_0^1 x^2(1+2x^3)^5\,dx$ with $u = 1 + 2x^3$, $x^2\,dx = \frac{1}{6}du$, limits $1 \to 3$: $\frac{1}{6}\int_1^3 u^5\,du = \frac{729 - 1}{36} = \frac{182}{9}$.

Common mistakes

• Not fixing the constant: if $du = 6x\,dx$ but you only have $x\,dx$, use $x\,dx = \frac{1}{6}du$.

• Keeping the old $x$-limits after switching to $u$.

• Forgetting to substitute back to $x$ in an indefinite integral.

2.6 Integrals of Exponential and Logarithmic Functions

Big idea

• $e^x$ is its own antiderivative. For other bases, divide by $\ln a$.

• $\int \frac{1}{x}\,dx = \ln|x|$ fills the gap in the power rule ($n = -1$).

• Big pattern: (derivative of bottom)/(bottom) integrates to $\ln|\text{bottom}|$.

• These show up in growth models (bacteria, price-demand): integrate the rate to get the change.

Formulas

• $\int e^x\,dx = e^x + C$, $\int a^x\,dx = \frac{a^x}{\ln a} + C$

• $\int \frac{1}{x}\,dx = \ln|x| + C$, $\int \frac{g'(x)}{g(x)}\,dx = \ln|g(x)| + C$

• $\int \ln x\,dx = x\ln x - x + C$, $\int \tan x\,dx = \ln|\sec x| + C$

Worked example

• Exponentials: $\int_0^{\ln 2} e^{3x}\,dx = \left.\frac{1}{3}e^{3x}\right|_0^{\ln 2} = \frac{1}{3}(8 - 1) = \frac{7}{3}$; and $\int 2^x\,dx = \frac{2^x}{\ln 2} + C$.

• Logarithms: $\int \frac{x}{x^2+1}\,dx$, $u = x^2 + 1$: $\frac{1}{2}\int \frac{du}{u} = \frac{1}{2}\ln(x^2+1) + C$ (no absolute value since $x^2 + 1 > 0$).

Common mistakes

• $\int a^x\,dx \ne \frac{a^{x+1}}{x+1}$; the power rule is for $x^n$, not $a^x$.

• Dropping the absolute value in $\ln|x|$; forgetting the $\frac{1}{k}$ in $\int e^{kx}\,dx = \frac{1}{k}e^{kx} + C$.

2.7 Integrals Resulting in Inverse Trig Functions

Big idea

• Three shapes in the bottom: $\sqrt{a^2 - u^2}$ gives arcsine; $a^2 + u^2$ gives arctangent; $u\sqrt{u^2 - a^2}$ gives arcsecant.

• Often a quick $u$-sub comes first ($u = 2x$, $u = x^2$, $u = e^x$).

Formulas

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

• $\int \frac{du}{a^2 + u^2} = \frac{1}{a}\tan^{-1}\left(\frac{u}{a}\right) + C$

• $\int \frac{du}{u\sqrt{u^2 - a^2}} = \frac{1}{a}\sec^{-1}\left(\frac{|u|}{a}\right) + C$

Worked example

• Arctangent with $u$-sub: $\int \frac{dx}{1+4x^2}$, $u = 2x$, $du = 2\,dx$: $\frac{1}{2}\int \frac{du}{1+u^2} = \frac{1}{2}\tan^{-1}(2x) + C$.

• Arcsine, definite: $\int_0^{\sqrt{3}} \frac{dx}{\sqrt{4 - x^2}} = \left.\sin^{-1}\left(\frac{x}{2}\right)\right|_0^{\sqrt{3}} = \sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$.

Common mistakes

• Forgetting the $\frac{1}{a}$ in front of arctan and arcsec (arcsine has none).

• Using $a^2$ instead of $a$: in $\frac{1}{9+x^2}$, $a = 3$. Using degrees; inverse trig answers are in radians.

How the pieces connect

• 2.1 builds Riemann sums; 2.2 takes $n \to \infty$ to define the definite integral as exact net signed area.

• 2.3 (FTC) replaces the limit with an antiderivative: $\int_a^b f = F(b) - F(a)$. The MVT for integrals comes from the average value in 2.2.

• 2.4 supplies the basic antiderivatives FTC 2 needs and reads the integral as net change (displacement vs. total distance), with even/odd shortcuts on $[-a,a]$.

• 2.5 (substitution) is the chain rule backwards; a quick $u$-sub usually sets up the $\ln$, exponential, or inverse-trig pattern in 2.6 and 2.7.

• Decision flow: (1) basic formula? use it; (2) inside function whose derivative is there? $u$-sub; (3) (derivative of bottom)/(bottom)? $\ln|\text{bottom}|$; (4) $\sqrt{a^2-u^2}$, $a^2+u^2$, or $u\sqrt{u^2-a^2}$ in the bottom? inverse trig; (5) symmetric interval? check even/odd.

Checklist

• 2.1: I can use the sum formulas; I can compute $L_n$ and $R_n$.

• 2.2: Net vs. total area; geometry integrals; average value.

• 2.3: FTC 1 (with chain rule); FTC 2; MVT for integrals.

• 2.4: Net change; total distance; even/odd.

• 2.5: Indefinite $u$-sub; definite $u$-sub with new limits.

• 2.6: $e^x$ and $a^x$ integrals; $\ln$ pattern.

• 2.7: Arcsine; arctangent; arcsecant.