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Lesson 82 — Definite integral and oriented area

Riemann sum as a limit. Definite integral as oriented area under the graph. Properties: linearity, additivity, monotonicity. Mean Value Theorem for Integrals.

Used in: 3rd year of high school (17 years old) · Equiv. Math II Japanese ch. 6 · Equiv. Grade 12 German Integral

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

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Rigorous notation, full derivation, hypotheses

Rigorous definition

Riemann sum

Definition· Riemann integral

Let $f: [a,b] \to \mathbb{R}$ be bounded. A partition $P$ of $[a,b]$ is a set $\{a = x_0 < x_1 < \cdots < x_n = b\}$ . The norm of the partition is $\|P\| = \max_i \Delta x_i$ where $\Delta x_i = x_i - x_{i-1}$ .

Choosing sample points $x_i^* \in [x_{i-1}, x_i]$ , the Riemann sum is:

$S(f, P) = \sum_{i=1}^n f(x_i^*)\, \Delta x_i.$

$f$ is integrable on $[a,b]$ if there exists a limit $L$ such that for every $\varepsilon > 0$ there exists $\delta > 0$ with $|S(f,P) - L| < \varepsilon$ whenever $\|P\| < \delta$ , regardless of the choice of $x_i^*$ . We write $\int_a^b f(x)\, dx = L$ .

"The definite integral is formally the limit of Riemann sums when the norm of the partition tends to zero." — OpenStax Calculus Vol. 1, §5.2

Darboux sums

Equivalent definition via lower and upper sums:

$L(f, P) = \sum_{i=1}^n \Bigl(\inf_{[x_{i-1}, x_i]} f\Bigr) \Delta x_i, \qquad U(f, P) = \sum_{i=1}^n \Bigl(\sup_{[x_{i-1}, x_i]} f\Bigr) \Delta x_i.$

$f$ is integrable $\iff \sup_P L(f,P) = \inf_P U(f,P)$ .

Integrability criterion

Properties

Definition· Properties of the definite integral

For $f$ , $g$ integrable on $[a, b]$ and $\alpha, \beta, c \in \mathbb{R}$ with $a \leq c \leq b$ :

Linearity: $\int_a^b (\alpha f + \beta g) = \alpha \int_a^b f + \beta \int_a^b g$
Additivity: $\int_a^b f = \int_a^c f + \int_c^b f$
Reversal of limits: $\int_b^a f = -\int_a^b f$
Zero integral: $\int_a^a f = 0$
Monotonicity: $f \leq g$ on $[a,b] \Rightarrow \int_a^b f \leq \int_a^b g$
Triangle inequality: $\left|\int_a^b f\right| \leq \int_a^b |f|$

Six Riemann rectangles approximating the integral. As $n \to \infty$ and $\|P\| \to 0$ , the sum converges to the exact area.

Mean Value Theorem for Integrals

Solved examples

Example— 1· Riemann sum with n = 4 (basic)

Problem. Estimate $\int_0^2 x^2\, dx$ using the right Riemann sum with $n = 4$ subintervals.

Strategy. Divide $[0, 2]$ into 4 equal parts, take the right point of each part, calculate the sum.

Solution. $\Delta x = 2/4 = 0{,}5$ . Right points: $x_1^* = 0{,}5$ , $x_2^* = 1$ , $x_3^* = 1{,}5$ , $x_4^* = 2$ .

$S_4 = f(0{,}5) + f(1) + f(1{,}5) + f(2)) \cdot 0{,}5 = (0{,}25 + 1 + 2{,}25 + 4) \cdot 0{,}5 = 7{,}5 \cdot 0{,}5 = 3{,}75.$

(The exact value by the Fundamental Theorem of Calculus is $\int_0^2 x^2\, dx = 8/3 \approx 2{,}67$ . The right sum overestimated because $x^2$ is increasing.)

Verification. The sum with large $n$ converges to $8/3$ : confirms that $3{,}75$ is an overestimate.

Source. APEX Calculus, §5.2, Example 5.2.4 — CC-BY-NC.

Example— 2· Geometric interpretation of integral with negative function (basic)

Problem. Interpret geometrically $\int_0^\pi \sin x\, dx$ and $\int_\pi^{2\pi} \sin x\, dx$ .

Strategy. $\sin x \geq 0$ on $[0, \pi]$ and $\sin x \leq 0$ on $[\pi, 2\pi]$ . The definite integral measures oriented area.

Solution.

$\int_0^\pi \sin x\, dx = [-\cos x]_0^\pi = -\cos\pi + \cos 0 = 1 + 1 = 2$ (area above the axis: $+2$ ).
$\int_\pi^{2\pi} \sin x\, dx = [-\cos x]_\pi^{2\pi} = -\cos 2\pi + \cos\pi = -1 - 1 = -2$ (area below the axis: contribution $-2$ ).

So $\int_0^{2\pi} \sin x\, dx = 2 + (-2) = 0$ . The geometric area total is $2 + 2 = 4$ , but the integral is zero.

Verification. Additivity: $\int_0^\pi + \int_\pi^{2\pi} = 2 - 2 = 0 = \int_0^{2\pi} \sin x\, dx$ . Check.

Source. OpenStax Calculus Vol. 1, §5.2, Example 5.18 — CC-BY-NC-SA.

Example— 4· Estimation by bound (intermediate)

Problem. Show that $2 \leq \int_0^2 \sqrt{x+1}\, dx \leq 2\sqrt{3}$ .

Strategy. Use monotonicity: bound $\sqrt{x+1}$ on $[0, 2]$ from below and above.

Solution. On $[0, 2]$ : minimum of $\sqrt{x+1}$ at $x = 0$ is $\sqrt{1} = 1$ ; maximum at $x = 2$ is $\sqrt{3}$ .

By monotonicity: $\int_0^2 1\, dx \leq \int_0^2 \sqrt{x+1}\, dx \leq \int_0^2 \sqrt{3}\, dx$ , i.e. $2 \leq \int \leq 2\sqrt{3}$ .

Verification. The exact value by the Fundamental Theorem of Calculus: $\int_0^2 \sqrt{x+1}\, dx = \left[\frac{2}{3}(x+1)^{3/2}\right]_0^2 = \frac{2}{3}(3^{3/2} - 1) = \frac{2}{3}(3\sqrt{3} - 1) \approx 2{,}80$ , which is between 2 and $2\sqrt{3} \approx 3{,}46$ . Check.

Source. Active Calculus, §4.2, Exercise 4.2.3 — CC-BY-NC-SA.

Example— 5· Mean value of continuous function (challenge)

Problem. Calculate the mean value of $f(x) = x^2$ on the interval $[1, 4]$ .

Strategy. Use the mean value formula: $f_{\text{mean}} = \frac{1}{b-a}\int_a^b f(x)\, dx$ .

Solution. $f_{\text{mean}} = \frac{1}{4-1}\int_1^4 x^2\, dx = \frac{1}{3}\left[\frac{x^3}{3}\right]_1^4 = \frac{1}{3}\left(\frac{64}{3} - \frac{1}{3}\right) = \frac{1}{3} \cdot 21 = 7.$

By the Mean Value Theorem for Integrals, there exists $c \in (1, 4)$ such that $f(c) = 7$ , i.e. $c^2 = 7$ , so $c = \sqrt{7} \approx 2{,}65 \in (1, 4)$ . Check.

Verification. $\sqrt{7} \approx 2{,}65$ is within the interval $(1, 4)$ , as guaranteed by the MVTI. Check.

Source. APEX Calculus, §5.3, Example 5.3.4 — CC-BY-NC.

Exercise list

30 exercises · 7 with worked solution (25%)

Application 20Understanding 2Modeling 4Challenge 3Proof 1

Ex. 82.1Application
Estimate $\int_0^4 x^2\, dx$ using the right Riemann sum with $n = 4$ and $\Delta x = 1$ .
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Ex. 82.2Application
Estimate $\int_0^4 x^2\, dx$ using the left Riemann sum with $n = 4$ and $\Delta x = 1$ .
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Ex. 82.3Application
Calculate $\int_0^3 (2x + 1)\, dx$ .
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Ex. 82.4Application
Calculate $\int_1^4 3x^2\, dx$ .
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Ex. 82.5ApplicationAnswer key
Calculate $\int_0^\pi \cos x\, dx$ and interpret the result geometrically.
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Ex. 82.6Application
Calculate $\int_0^1 e^x\, dx$ .
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Ex. 82.7Application
Calculate $\int_1^e \frac{1}{x}\, dx$ .
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Ex. 82.8Application
Calculate $\int_0^2 (3x^2 - 4x + 1)\, dx$ .
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Ex. 82.9Application
Calculate $\int_0^{\pi/2} \sin x\, dx$ .
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Ex. 82.10Application
Calculate $\int_{-1}^2 x^3\, dx$ .
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Ex. 82.11Application
Given that $\int_0^2 f(x)\, dx = 3$ and $\int_2^5 f(x)\, dx = -4$ , calculate $\int_0^5 f(x)\, dx$ .
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Ex. 82.12ApplicationAnswer key
Given that $\int_1^3 f(x)\, dx = 5$ and $\int_1^3 g(x)\, dx = 7$ , calculate $\int_1^3 (4f(x) - 2g(x))\, dx$ .
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Ex. 82.13Application
If $\int_2^5 f(x)\, dx = -4$ , what is $\int_5^2 f(x)\, dx$ ?
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Ex. 82.14ApplicationAnswer key
Calculate $\int_0^4 \sqrt{x}\, dx$ .
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Ex. 82.15Application
Calculate $\int_0^{\pi/4} \sec^2 x\, dx$ .
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Ex. 82.16Understanding
Without calculating, what is the sign of $\int_{-\pi}^\pi \sin x\, dx$ ?
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Ex. 82.17Understanding
Which statement about $\int_a^b f(x)\, dx$ is correct?
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Ex. 82.18ModelingAnswer key
A vehicle has velocity $v(t) = 3t^2 + 2$ m/s. What is the distance traveled from $t = 0$ to $t = 4$ s?
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Ex. 82.19ModelingAnswer key
The temperature of an industrial reactor varies as $T(t) = 2t + 1$ °C during the first 6 hours of operation. Calculate the average temperature in this period.
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Ex. 82.20ApplicationAnswer key
Given that $\int_1^5 f(x)\, dx = 10$ and $\int_3^5 f(x)\, dx = 4$ , calculate $\int_1^3 f(x)\, dx$ .
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Ex. 82.21Application
Calculate $\int_{-2}^2 x^3\, dx$ .
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Ex. 82.22Application
Calculate $\int_2^5 (4 - x)\, dx$ .
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Ex. 82.23Modeling
Calculate the total geometric area (always positive) bounded by $y = \sin x$ and the $x$ -axis on $[0, 2\pi]$ .
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Ex. 82.24Challenge
Use the monotonicity property to establish upper and lower bounds for $\int_0^1 (x^2 + 1)\, dx$ , without calculating.
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Ex. 82.25Challenge
Calculate the mean value of $f(x) = \sin x$ on $[0, \pi]$ and find the value of $c$ guaranteed by the Mean Value Theorem for Integrals.
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Ex. 82.26Application
Calculate $\int_0^{\pi/2} (\sin x + \cos x)\, dx$ .
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Ex. 82.27Application
Calculate $\int_0^2 (e^x - 1)\, dx$ .
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Ex. 82.28Challenge
Establish bounds for $\int_1^3 \sqrt{x}\, dx$ and then calculate the exact value.
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Ex. 82.29ModelingAnswer key
A variable force $F(x) = 10 - 2x$ N acts on an object that moves from $x = 0$ to $x = 3$ m. Calculate the work done ( $W = \int_0^3 F(x)\, dx$ ).
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Ex. 82.30Proof
Prove the reversal of limits property: $\int_b^a f(x)\, dx = -\int_a^b f(x)\, dx$ .

Sources

Active Calculus — Boelkins · §4.2 · CC-BY-NC-SA. Intuitive construction of Riemann sums, graphing activities.
APEX Calculus — Hartman et al. · §5.2–5.3 · CC-BY-NC. Formal definition, properties, numerical examples.
OpenStax Calculus Volume 1 · §5.2 · CC-BY-NC-SA. Integrability criterion, properties, examples with positive and negative functions.