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Lesson 90 — Term 9 Consolidation (integral calculus)

Integrative workshop: antiderivative, definite integral, FTC, substitution, parts, partial fractions, trig integrals, area and volume.

Used in: 3º year of High School (17–18 years old) · Equiv. Japanese Math III (ch. 5–6) · Equiv. German Leistungskurs Integralrechnung II

\int_a^b f(x)\,dx = F(b) - F(a), \quad \int u\,dv = uv - \int v\,du, \quad A = \int_a^b [f(x)-g(x)]\,dx

Choose your door

Rigorous notation, full derivation, hypotheses

Rigorous synthesis of the term

Conceptual map of integral calculus

Decision tree — "Which technique to use?"

Flow for integrating $\int f\,dx$ . Follow top to bottom; apply the first technique that fits.

Quick table of fundamental antiderivatives

Canonical applications

Definition· Area and volume by integration

Area between curves. If $f(x) \geq g(x)$ on $[a, b]$ :

$A = \int_a^b [f(x) - g(x)]\,dx$

Volume by disks/washers. Revolution of $y = f(x)$ about the $x$ -axis:

$V = \pi \int_a^b [f(x)]^2\,dx$

For washer (region between $f$ and $g$ , $f \geq g \geq 0$ ):

$V = \pi \int_a^b \bigl([f(x)]^2 - [g(x)]^2\bigr)\,dx$

Volume by cylindrical shells. Revolution of $y = f(x) \geq 0$ about the $y$ -axis:

$V = 2\pi \int_a^b x f(x)\,dx$

"The method of cylindrical shells is preferable when the region rotates about an axis parallel to the axis of integration." — APEX Calculus §7.3, p. 359

Worked examples

Example— 1· FTC2 applied with substitution in limits (application)

Problem: Compute $\displaystyle\int_0^1 2x e^{x^2}\,dx$ .

Strategy: The integrand has the form $f(g(x)) \cdot g'(x)$ with $g(x) = x^2$ and $g'(x) = 2x$ . Apply substitution $u = x^2$ and swap integration limits directly in $u$ .

Solution:

Let $u = x^2$ , then $du = 2x\,dx$ .

Limits: $x = 0 \Rightarrow u = 0$ ; $x = 1 \Rightarrow u = 1$ .

$\int_0^1 2x e^{x^2}\,dx = \int_0^1 e^u\,du = \bigl[e^u\bigr]_0^1 = e^1 - e^0 = e - 1$

Verification: The antiderivative of $2x e^{x^2}$ is $e^{x^2}$ (differentiating: $(e^{x^2})' = e^{x^2} \cdot 2x$ ✓). Evaluating: $e^{1^2} - e^{0^2} = e - 1 \approx 1.718$ .

Source. Active Calculus §5.3, Activity 5.3.2 — CC-BY-NC-SA.

Example— 2· Iterated integration by parts — xe^x (intermediate)

Problem: Compute $\displaystyle\int x^2 e^x\,dx$ .

Strategy: Product of polynomial by exponential, no chain relation between them. Use integration by parts with LIATE: $u = x^2$ (algebraic) and $dv = e^x\,dx$ (exponential). Will need to apply parts twice.

Solution:

1st application: $u = x^2$ , $du = 2x\,dx$ ; $dv = e^x\,dx$ , $v = e^x$ .

$\int x^2 e^x\,dx = x^2 e^x - \int 2x e^x\,dx$

2nd application (for $\int 2x e^x\,dx$ ): $u = 2x$ , $du = 2\,dx$ ; $dv = e^x\,dx$ , $v = e^x$ .

$\int 2x e^x\,dx = 2x e^x - \int 2 e^x\,dx = 2x e^x - 2e^x$

Final result:

$\int x^2 e^x\,dx = x^2 e^x - (2x e^x - 2e^x) + C = (x^2 - 2x + 2)e^x + C$

Verification: $\frac{d}{dx}\bigl[(x^2 - 2x + 2)e^x\bigr] = (2x - 2)e^x + (x^2 - 2x + 2)e^x = x^2 e^x$ ✓.

Source. APEX Calculus §8.1, Example 8.1.4 — CC-BY-NC.

Example— 3· Partial fractions — denominator with two linear factors (intermediate)

Problem: Compute $\displaystyle\int \frac{1}{x^2 - 1}\,dx$ .

Strategy: The integrand is rational with $\deg(\text{num}) < \deg(\text{den})$ . Factor the denominator: $x^2 - 1 = (x-1)(x+1)$ . Decompose into partial fractions.

Solution:

$\frac{1}{(x-1)(x+1)} = \frac{A}{x-1} + \frac{B}{x+1}$

Multiplying both sides by $(x-1)(x+1)$ : $1 = A(x+1) + B(x-1)$ .

$x = 1$ : $1 = 2A \Rightarrow A = 1/2$ . $x = -1$ : $1 = -2B \Rightarrow B = -1/2$ .

$\int \frac{1}{x^2 - 1}\,dx = \int \frac{1/2}{x-1}\,dx + \int \frac{-1/2}{x+1}\,dx = \frac{1}{2}\ln\lvert x - 1\rvert - \frac{1}{2}\ln\lvert x + 1\rvert + C$

$= \frac{1}{2}\ln\left\lvert\frac{x-1}{x+1}\right\rvert + C$

Verification: Differentiating: $\frac{1}{2} \cdot \frac{(x+1) - (x-1)}{(x-1)(x+1)} = \frac{1}{2} \cdot \frac{2}{x^2-1} = \frac{1}{x^2-1}$ ✓.

Source. OpenStax Calculus Vol. 2 §3.4, Example 3.4.1 — CC-BY-NC-SA.

Example— 4· Area between parabola and line (application)

Problem: Compute the area of the region bounded by $f(x) = x^2$ and $g(x) = x + 2$ .

Strategy: Find the intersections (points where $f = g$ ), determine which curve is above in each subinterval, and integrate the difference.

Solution:

Intersections: $x^2 = x + 2 \Rightarrow x^2 - x - 2 = 0 \Rightarrow (x-2)(x+1) = 0 \Rightarrow x = -1$ or $x = 2$ .

Which is "on top"? For $x \in (-1, 2)$ : $g(0) = 2 > f(0) = 0$ , so $g \geq f$ .

$A = \int_{-1}^2 [g(x) - f(x)]\,dx = \int_{-1}^2 (x + 2 - x^2)\,dx$

$= \left[\frac{x^2}{2} + 2x - \frac{x^3}{3}\right]_{-1}^2 = \left(2 + 4 - \frac{8}{3}\right) - \left(\frac{1}{2} - 2 + \frac{1}{3}\right) = \frac{10}{3} - \left(-\frac{7}{6}\right) = \frac{10}{3} + \frac{7}{6} = \frac{27}{6} = \frac{9}{2}$

Verification: $9/2 = 4.5$ square units. The parabola and line form a tear-drop-shaped region — reasonable result for an interval of length 3.

Source. Active Calculus §6.1, Activity 6.1.2 — CC-BY-NC-SA.

Example— 5· Volume of revolution by shell method (advanced)

Problem: Compute the volume of the solid generated by rotating the region bounded by $y = \ln x$ , $y = 0$ and $x = e$ about the $y$ -axis.

Strategy: The region is to the right of the $y$ -axis. Rotation about the $y$ -axis with integration in $x$ suggests the cylindrical shell method: $V = 2\pi \int_a^b x f(x)\,dx$ .

Solution:

The solid goes from $x = 1$ (since $\ln 1 = 0$ ) to $x = e$ .

$V = 2\pi \int_1^e x \ln x\,dx$

Apply integration by parts: $u = \ln x$ , $dv = x\,dx$ ; $du = dx/x$ , $v = x^2/2$ .

$\int x \ln x\,dx = \frac{x^2}{2}\ln x - \int \frac{x^2}{2} \cdot \frac{1}{x}\,dx = \frac{x^2}{2}\ln x - \frac{x^2}{4} + C$

Evaluating:

$\int_1^e x \ln x\,dx = \left[\frac{x^2}{2}\ln x - \frac{x^2}{4}\right]_1^e = \left(\frac{e^2}{2} - \frac{e^2}{4}\right) - \left(0 - \frac{1}{4}\right) = \frac{e^2}{4} + \frac{1}{4} = \frac{e^2 + 1}{4}$

$V = 2\pi \cdot \frac{e^2 + 1}{4} = \frac{\pi(e^2 + 1)}{2} \approx \frac{\pi \cdot 8.389}{2} \approx 13.18$

Verification: For $x \in [1, e]$ , the shell has radius $x$ and height $\ln x$ . The result has correct dimension (unit³). $\approx 13.18$ is reasonable for this region.

Source. OpenStax Calculus Vol. 2 §2.3, Example 2.3.4 — CC-BY-NC-SA.

Exercise list

40 exercises · 10 with worked solution (25%)

Application 25Understanding 4Modeling 3Challenge 4Proof 4

Ex. 90.1Application
Compute $\int (6x^2 - 5)\,dx$ .
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Ex. 90.2Application
Compute $\displaystyle\int_0^1 (3x^2 - 2x)\,dx$ .
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Ex. 90.3ApplicationAnswer key
Compute $\displaystyle\int_0^{\pi/2} \cos x\,dx$ .
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Ex. 90.4Application
Compute $\displaystyle\int_1^e \frac{1}{x}\,dx$ .
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Ex. 90.5ApplicationAnswer key
Let $G(x) = \displaystyle\int_1^x (t^2 - 3t)\,dt$ . Determine $G'(x)$ .
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Ex. 90.6Understanding
What is the general antiderivative of $f(x) = 3x^2$ ?
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Ex. 90.7Application
Compute $\displaystyle\int_0^1 \frac{1}{1 + x^2}\,dx$ .
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Ex. 90.8ProofAnswer key
Prove FTC2 assuming FTC1 as a starting point.
Ex. 90.9Application
Compute $\int (x^3 + 1)^4 \cdot x^2\,dx$ .
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Ex. 90.10ApplicationAnswer key
Compute $\int \frac{\cos x}{\sin x}\,dx$ .
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Ex. 90.11Application
Compute $\displaystyle\int_0^1 x e^{x^2}\,dx$ .
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Ex. 90.12Application
Compute $\int \sin(e^x)\,e^x\,dx$ .
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Ex. 90.13Application
Compute $\int \frac{1}{\sqrt{x + 3}}\,dx$ .
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Ex. 90.14Understanding
For $\int 2x(x^2 + 1)^3\,dx$ , which substitution is appropriate?
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Ex. 90.15Application
Compute $\int \sin^3 x\,dx$ .
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Ex. 90.16Application
Compute $\int \sqrt{1 - x^2}\,dx$ using trigonometric substitution $x = \sin\theta$ .
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Ex. 90.17Application
Compute $\int x e^x\,dx$ .
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Ex. 90.18ApplicationAnswer key
Compute $\int \ln x\,dx$ .
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Ex. 90.19Application
Compute $\int \arctan x\,dx$ .
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Ex. 90.20Application
Compute $\displaystyle\int_1^e \ln x\,dx$ .
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Ex. 90.21Application
Compute $\displaystyle\int \frac{1}{x^2 - x - 2}\,dx$ . (Factor the denominator first.)
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Ex. 90.22Application
Compute $\displaystyle\int \frac{x}{x^2 - 4}\,dx$ .
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Ex. 90.23Application
Compute $\int x^2 \cos x\,dx$ .
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Ex. 90.24Challenge
Compute $\int e^x \sin x\,dx$ using the trick of the integral that "returns to itself".
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Ex. 90.25Proof
Prove the integration by parts formula $\int u\,dv = uv - \int v\,du$ from the product rule.
Ex. 90.26Application
Compute the area of the region bounded by $y = x + 2$ and $y = x^2$ .
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Ex. 90.27Application
Compute the area between $y = \sin x$ and $y = \cos x$ on $[0, \pi]$ .
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Ex. 90.28Application
Compute the volume of the solid generated by revolving $y = e^{-x}$ , $x \in [0, 2]$ , about the $x$ -axis.
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Ex. 90.29Application
Compute the volume of the solid between $y = x$ and $y = x^2$ ( $x \in [0,1]$ ) rotated about the $x$ -axis.
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Ex. 90.30Application
Compute the volume of the solid generated by $y = 1 - x^2$ , $x \in [0, 1]$ , rotated about the $y$ -axis by the method of cylindrical shells.
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Ex. 90.31ModelingAnswer key
Compute the volume generated by $y = \ln x$ , $x \in [1, e]$ , rotated about the $y$ -axis (shells). Combine by parts and substitution.
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Ex. 90.32Understanding
When is it preferable to use cylindrical shells instead of disks to compute volume?
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Ex. 90.33Modeling
Compute the area between $y = x$ and $y = x^3$ on $[-1, 1]$ .
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Ex. 90.34ModelingAnswer key
The force on a piston is $F(x) = xe^{-x}$ N for $x \in [0, 5]$ m. Compute the work done.
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Ex. 90.35Challenge
Gabriel's trumpet: $y = 1/x$ on $[1, \infty)$ rotated about the $x$ -axis. Show that the volume is $\pi$ and discuss why the surface area is infinite.
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Ex. 90.36ProofAnswer key
Show that $\displaystyle\int_0^1 (1-x)^n\,dx = \dfrac{1}{n+1}$ for $n \in \mathbb{N}$ using substitution.
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Ex. 90.37Proof
Show that $\displaystyle\int_0^{\pi} \sin^2(nx)\,dx = \dfrac{\pi}{2}$ for all $n \in \mathbb{N}$ .
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Ex. 90.38ChallengeAnswer key
Show (by sketching the idea with polars) that $\displaystyle\int_0^\infty e^{-x^2}\,dx = \dfrac{\sqrt{\pi}}{2}$ .
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Ex. 90.39Understanding
The integral $\displaystyle\int_1^\infty \frac{1}{x}\,dx$ : does it converge or diverge?
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Ex. 90.40ChallengeAnswer key
Compute $\displaystyle\int_0^1 x \arctan x\,dx$ .
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Sources

Active Calculus — Matt Boelkins · 2024 · ed. 2.0 · EN · CC-BY-NC-SA · §4.1–4.4, §5.1–6.2. Primary source.
Calculus Volume 2 — OpenStax (Herman et al.) · 2016 · EN · CC-BY-NC-SA · ch. 1–3.
APEX Calculus — Hartman, Heinold, Siemers, Chalishajar · 2024 · v5 · EN · CC-BY-NC · ch. 5–8.

$f(x)$	$\int f(x)\,dx$
$x^n$ ( $n \neq -1$ )	$\dfrac{x^{n+1}}{n+1} + C$
$x^{-1}$	$\ln\lvert x\rvert + C$
$e^x$	$e^x + C$
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$
$\dfrac{1}{\sqrt{1-x^2}}$	$\arcsin x + C$
$\dfrac{1}{1+x^2}$	$\arctan x + C$