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Home/Calculus/Chapter 6/Properties of Definite Integrals

CALC-6.2

3-4 hours

Properties of Definite Integrals

Master the fundamental properties that make definite integrals a powerful tool: linearity, comparison, interval additivity, and more.

Learning Objectives

Prove linearity of the definite integral

Establish interval additivity property

Apply comparison theorems and estimation

Prove absolute value integrability

Understand product integrability

Apply Cauchy-Schwarz inequality for integrals

1. Linearity

Theorem 6.9: Linearity of Definite Integrals

If $f, g \in R[a, b]$ and $\alpha, \beta \in \mathbb{R}$ , then $\alpha f + \beta g \in R[a, b]$ and:

\int_a^b [\alpha f(x) + \beta g(x)]\,dx = \alpha \int_a^b f(x)\,dx + \beta \int_a^b g(x)\,dx

Proof of Theorem 6.9:

For any partition $\Delta$ and sample points $\{\xi\}$ :

S_\Delta(\alpha f + \beta g, \xi) = \sum_{k=1}^n [\alpha f(\xi_k) + \beta g(\xi_k)]\Delta x_k

= \alpha \sum_{k=1}^n f(\xi_k)\Delta x_k + \beta \sum_{k=1}^n g(\xi_k)\Delta x_k = \alpha S_\Delta(f, \xi) + \beta S_\Delta(g, \xi)

Taking $\|\Delta\| \to 0$ gives the result.

∎

Example 6.7: Using Linearity

Compute: $\int_0^1 (3x^2 - 2x + 5)\,dx$

= 3\int_0^1 x^2\,dx - 2\int_0^1 x\,dx + 5\int_0^1 1\,dx

= 3 \cdot \frac{1}{3} - 2 \cdot \frac{1}{2} + 5 \cdot 1 = 1 - 1 + 5 = 5

2. Comparison Theorems

Theorem 6.10: Order Preservation

If $f, g \in R[a, b]$ and $f(x) \leq g(x)$ for all $x \in [a, b]$ , then:

\int_a^b f(x)\,dx \leq \int_a^b g(x)\,dx

Corollary 6.2: Non-negativity

If $f \in R[a, b]$ and $f(x) \geq 0$ on $[a, b]$ , then:

\int_a^b f(x)\,dx \geq 0

Corollary 6.3: Estimation Theorem

If $m \leq f(x) \leq M$ on $[a, b]$ , then:

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

Example 6.8: Estimating an Integral

Estimate: $\int_0^1 \sqrt{1 + x^3}\,dx$

On $[0, 1]$ : $0 \leq x^3 \leq 1$ , so $1 \leq 1 + x^3 \leq 2$ .

Thus $1 \leq \sqrt{1 + x^3} \leq \sqrt{2}$ .

1 \cdot 1 \leq \int_0^1 \sqrt{1 + x^3}\,dx \leq \sqrt{2} \cdot 1

Result: $1 \leq I \leq \sqrt{2} \approx 1.414$

3. Interval Additivity

Theorem 6.11: Interval Additivity

For $a < c < b$ :

f \in R[a, b] \Leftrightarrow f \in R[a, c] \cap R[c, b]

In this case:

\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx

Remark 6.3

This property extends to any finite number of subintervals. Combined with our conventions, it holds for any ordering of $a, b, c$ .

4. Absolute Value Integrability

Theorem 6.12: Absolute Value Integrability

If $f \in R[a, b]$ , then $|f| \in R[a, b]$ , and:

\left| \int_a^b f(x)\,dx \right| \leq \int_a^b |f(x)|\,dx

Proof of Theorem 6.12:

For integrability of $|f|$ : Since $||f(x)| - |f(y)|| \leq |f(x) - f(y)|$ , the oscillation of $|f|$ on any interval is ≤ that of $f$ .

For the inequality: $-|f| \leq f \leq |f|$ implies $-\int|f| \leq \int f \leq \int|f|$ .

∎

Remark 6.4: Converse is False

Warning: $|f| \in R[a,b]$ does NOT imply $f \in R[a,b]$ !

Example: $f(x) = 1$ if $x \in \mathbb{Q}$ , $f(x) = -1$ if $x \notin \mathbb{Q}$ .

Then $|f| \equiv 1$ is integrable, but $f$ is not.

5. Product and Special Inequalities

Theorem 6.13: Product Integrability

If $f, g \in R[a, b]$ , then $f \cdot g \in R[a, b]$ .

Theorem 6.14: Cauchy-Schwarz Inequality

For $f, g \in R[a, b]$ :

\left( \int_a^b |f(x)g(x)|\,dx \right)^2 \leq \int_a^b f^2(x)\,dx \cdot \int_a^b g^2(x)\,dx

Example 6.9: Applying Cauchy-Schwarz

Prove: $\left(\int_0^1 \sqrt{x}\,dx\right)^2 \leq \int_0^1 x\,dx \cdot \int_0^1 1\,dx$

Taking $f(x) = \sqrt{x}$ , $g(x) = 1$ :

\left(\frac{2}{3}\right)^2 \leq \frac{1}{2} \cdot 1

\frac{4}{9} \leq \frac{1}{2} \quad \checkmark

Theorem 6.15: Riemann-Lebesgue Lemma

If $f \in R[a, b]$ , then:

\lim_{\lambda \to +\infty} \int_a^b f(x) \sin(\lambda x)\,dx = 0

Similarly for $\cos(\lambda x)$ . This is fundamental in Fourier analysis.

6. Important Examples

Example 6.10: Integral Inequality

Problem: If $f \in C[0, a]$ , $f(0) = 0$ , and $f'(x) \leq M$ , prove $\int_0^a f(x)\,dx \leq \frac{Ma^2}{2}$ .

Solution:

By the Mean Value Theorem: $f(x) = f(x) - f(0) = f'(\xi)x \leq Mx$

\int_0^a f(x)\,dx \leq \int_0^a Mx\,dx = M \cdot \frac{a^2}{2}

Example 6.11: Zero Integral Implies Zero Function

Theorem: If $f \in C[a, b]$ , $f(x) \geq 0$ , and $\int_a^b f(x)\,dx = 0$ , then $f \equiv 0$ .

Proof:

Suppose $f(x_0) > 0$ for some $x_0$ . By continuity, $f(x) > \frac{f(x_0)}{2}$ on some $[x_0-\delta, x_0+\delta]$ .

\int_a^b f \geq \int_{x_0-\delta}^{x_0+\delta} f > \frac{f(x_0)}{2} \cdot 2\delta > 0

Contradiction! So $f \equiv 0$ .

7. Additional Examples

Example 6.12: Integral of Odd Function over Symmetric Interval

Theorem: If $f$ is odd ( $f(-x) = -f(x)$ ) and integrable on $[-a, a]$ , then:

\int_{-a}^a f(x)\,dx = 0

Proof:

By interval additivity and substitution $u = -x$ :

\int_{-a}^0 f(x)\,dx = -\int_a^0 f(-u)\,du = \int_0^a f(-u)\,du = -\int_0^a f(u)\,du

Thus $\int_{-a}^a f = \int_{-a}^0 f + \int_0^a f = -\int_0^a f + \int_0^a f = 0$ .

Example 6.13: Integral of Even Function

Theorem: If $f$ is even ( $f(-x) = f(x)$ ) and integrable on $[-a, a]$ , then:

\int_{-a}^a f(x)\,dx = 2\int_0^a f(x)\,dx

This often simplifies computation of integrals over symmetric intervals.

Example 6.14: Applying Symmetry

Compute: $\int_{-1}^1 (x^3 + 2x^2 + 3x + 4)\,dx$

Solution:

Split into odd and even parts:

$x^3$ and $3x$ are odd → contribute 0
$2x^2$ and $4$ are even → double the integral on $[0,1]$

= 2\int_0^1 (2x^2 + 4)\,dx = 2\left[\frac{2x^3}{3} + 4x\right]_0^1 = 2\left(\frac{2}{3} + 4\right) = \frac{28}{3}

Example 6.15: Squeeze Theorem for Integrals

Problem: Evaluate $\lim_{n \to \infty} \int_0^1 \frac{x^n}{1+x}\,dx$ .

Solution:

On $[0, 1]$ : $0 \leq \frac{x^n}{1+x} \leq \frac{x^n}{1} = x^n$ .

By comparison:

0 \leq \int_0^1 \frac{x^n}{1+x}\,dx \leq \int_0^1 x^n\,dx = \frac{1}{n+1} \to 0

Answer: The limit is $0$ .

Example 6.16: Using Comparison to Evaluate Limits

Problem: Find $\lim_{n \to \infty} \int_0^{\pi/2} \sin^n x\,dx$ .

Solution:

For $x \in (0, \pi/2)$ : $0 < \sin x < 1$ , so $\sin^n x \to 0$ pointwise.

Moreover, $0 \leq \sin^n x \leq 1$ , and by dominated convergence (or direct estimation):

\int_0^{\pi/2} \sin^n x\,dx \leq \int_0^{\delta} 1\,dx + \int_\delta^{\pi/2} \sin^n \delta\,dx \to 0

Answer: The limit is $0$ .

8. Practice Problems

Problem Set A: Linearity and Comparison

1.Compute $\int_{-2}^2 (5x^4 - 3x^3 + x^2 - x + 2)\,dx$ using symmetry.
2.Show that $\int_0^{\pi/2} \sin x\,dx \leq \frac{\pi}{2}$ .
3.Prove: $\frac{\pi}{6} \leq \int_0^{\pi/4} \frac{dx}{\cos x} \leq \frac{\pi}{4\cos(\pi/4)}$ .
4.If $f \in C[0, 1]$ and $\int_0^1 x^n f(x)\,dx = 0$ for all $n \geq 0$ , prove $f \equiv 0$ .

Problem Set B: Absolute Value and Products

5.Show: $\left|\int_0^\pi \sin x \cos(nx)\,dx\right| \leq \frac{2}{n^2-1}$ for $n > 1$ .
6.Use Cauchy-Schwarz to prove $\left(\int_a^b f\right)^2 \leq (b-a)\int_a^b f^2$ .
7.Show that $\int_0^1 \sqrt{x}\sqrt{1-x}\,dx \leq \frac{1}{4}$ .
8.Prove the Hölder inequality: $\int|fg| \leq (\int|f|^p)^{1/p}(\int|g|^q)^{1/q}$ where $1/p + 1/q = 1$ .

Problem Set C: Applications

9.Prove: $\int_0^1 \frac{x^{100}}{1+x}\,dx < 0.01$ .
10.Show: $0.4 < \int_0^1 \frac{1}{\sqrt{1+x^4}}\,dx < 1$ .
11.Evaluate $\lim_{n \to \infty} \int_0^1 nx e^{-nx^2}\,dx$ .
12.If $f \in C[a,b]$ with $\int_a^b f = 0$ , show $f$ has a zero in $(a,b)$ (unless $f \equiv 0$ ).

Remark 6.5: Answers to Selected Problems

Problem 1: $2(5 \cdot \frac{32}{5} + \frac{8}{3} + 4) = \frac{296}{15}$

Problem 9: Use $x^{100} < x^{100} < \frac{1}{101}$

Problem 11: $\frac{1}{2}$

Problem 12: By IVT, $f$ takes positive and negative values

9. Deeper Insights

Why These Properties Matter

Linearity

Linearity makes the integral a linear functional, connecting calculus to linear algebra and functional analysis. This is the foundation of Fourier analysis and quantum mechanics.

Comparison

Comparison theorems let us bound integrals we cannot compute exactly. This is crucial in convergence tests for improper integrals and series.

Absolute Value

The triangle inequality $|\int f| \leq \int|f|$ is essential for proving convergence of integral series and for error estimates in numerical integration.

Cauchy-Schwarz

This inequality defines an inner product on $L^2$ functions, making them a Hilbert space. This structure underlies quantum mechanics and signal processing.

Theorem 6.16: Minkowski's Inequality

For $f, g \in R[a, b]$ and $p \geq 1$ :

\left(\int_a^b |f+g|^p\right)^{1/p} \leq \left(\int_a^b |f|^p\right)^{1/p} + \left(\int_a^b |g|^p\right)^{1/p}

This is the triangle inequality in $L^p$ space.

Remark 6.6: Connection to Norms

The integral defines various norms on function spaces:

$L^1$ norm

\|f\|_1 = \int|f|

$L^2$ norm

\|f\|_2 = \sqrt{\int f^2}

$L^\infty$ norm

\|f\|_\infty = \sup|f|

10. Historical Context

Augustin-Louis Cauchy (1789-1857)

First proved the comparison theorem rigorously. His work on complex analysis also used these properties to develop contour integration.

Hermann Schwarz (1843-1921)

Along with Cauchy, developed the Cauchy-Schwarz inequality, now fundamental to inner product spaces and quantum mechanics.

Hermann Minkowski (1864-1909)

Developed the Minkowski inequality and the geometry of numbers. His work connected integration theory to convex geometry.

11. Common Mistakes

❌ Wrong direction in triangle inequality

$|\int f| \leq \int|f|$ , NOT $|\int f| \geq \int|f|$ !

❌ Assuming $|f|$ integrable implies $f$ integrable

The converse is true! $f \in R \Rightarrow |f| \in R$ .

❌ Forgetting $(b-a)$ in estimation

If $m \leq f \leq M$ , then $m(b-a) \leq \int f \leq M(b-a)$ .

❌ Misusing Cauchy-Schwarz

Remember: $(\int fg)^2 \leq (\int f^2)(\int g^2)$ , not $\int f \cdot \int g$ .

12. Summary of Key Properties

Property	Formula	Conditions
Linearity	$\int(\alpha f + \beta g) = \alpha\int f + \beta\int g$	$f, g \in R[a,b]$
Comparison	$f \leq g \Rightarrow \int f \leq \int g$	$f, g \in R[a,b]$
Additivity	$\int_a^b f = \int_a^c f + \int_c^b f$	$a < c < b$
Estimation	$m(b-a) \leq \int f \leq M(b-a)$	$m \leq f \leq M$
Triangle	$\|\int f\| \leq \int\|f\|$	$f \in R[a,b]$
Cauchy-Schwarz	$(\int fg)^2 \leq \int f^2 \cdot \int g^2$	$f, g \in R[a,b]$

13. What's Next?

In the next section, we discover the Fundamental Theorem of Calculus—the profound connection between differentiation and integration:

Variable upper limit integrals:

F(x) = \int_a^x f(t)\,dt

FTC Part 1:

F'(x) = f(x)

Newton-Leibniz formula:

\int_a^b f = F(b) - F(a)

Applications to computing definite integrals

14. More Advanced Examples

Example 6.17: Integral of Periodic Function

Theorem: If $f$ has period $T$ , then for any $a$ :

\int_a^{a+T} f(x)\,dx = \int_0^T f(x)\,dx

Proof:

By substitution $u = x - a$ on the left, and using periodicity:

\int_a^{a+T} f(x)\,dx = \int_0^T f(u+a)\,du = \int_0^T f(u)\,du

Example 6.18: Bonnet's Mean Value Theorem

Theorem: If $f$ is monotonic and $g \in R[a,b]$ , then $\exists \xi \in [a,b]$ :

\int_a^b f(x)g(x)\,dx = f(a)\int_a^\xi g(x)\,dx + f(b)\int_\xi^b g(x)\,dx

This is useful for estimating products where one factor is monotonic.

Example 6.19: Chebyshev's Inequality

Theorem: If $f, g$ are both increasing (or both decreasing) on $[a,b]$ :

\int_a^b f(x)g(x)\,dx \geq \frac{1}{b-a}\int_a^b f(x)\,dx \cdot \int_a^b g(x)\,dx

Equality holds iff one of $f, g$ is constant.

Example 6.20: Jensen's Inequality for Integrals

Theorem: If $\phi$ is convex and $f \in R[a,b]$ :

\phi\left(\frac{1}{b-a}\int_a^b f(x)\,dx\right) \leq \frac{1}{b-a}\int_a^b \phi(f(x))\,dx

Example: $\phi(x) = x^2$ gives $(\bar{f})^2 \leq \overline{f^2}$ .

15. Study Tips

Master the Basics First

Linearity and comparison are used constantly. Make sure you can apply them automatically before moving to more complex properties like Cauchy-Schwarz.

Practice Estimation

When you can't compute an integral exactly, use bounds. Find $m \leq f \leq M$ and apply the estimation theorem.

Remember Direction of Inequalities

Triangle inequality: $|\int f| \leq \int|f|$ . The absolute value of the integral is LESS than the integral of absolute value.

Use Symmetry

For symmetric intervals $[-a, a]$ : odd functions integrate to 0, even functions double. This can save enormous computation.

16. Quick Reference Card

Key Formulas

Linearity

\int(\alpha f + \beta g) = \alpha\int f + \beta\int g

Additivity

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

Triangle

|\int f| \leq \int|f|

C-S

(\int fg)^2 \leq \int f^2 \int g^2

Symmetry Properties

Odd function:

\int_{-a}^a f(x)\,dx = 0

Even function:

\int_{-a}^a f(x)\,dx = 2\int_0^a f(x)\,dx

Reverse limits:

\int_b^a f = -\int_a^b f

Preservation

$f \in R \Rightarrow |f| \in R$
$f, g \in R \Rightarrow fg \in R$
$f \in R \Rightarrow f^2 \in R$

Comparison

$f \leq g \Rightarrow \int f \leq \int g$
$f \geq 0 \Rightarrow \int f \geq 0$
$m \leq f \leq M$ bounds

Important Limits

$\lim \int x^n \to 0$ as $n \to \infty$
Riemann-Lebesgue lemma
Dominated convergence

17. Extended Applications

Application to Probability

For a probability density function $f(x) \geq 0$ with $\int_{-\infty}^\infty f = 1$ :

Expected value: $E[X] = \int x f(x)\,dx$
Variance: $\text{Var}(X) = E[X^2] - (E[X])^2$
Cauchy-Schwarz gives: $E[XY]^2 \leq E[X^2]E[Y^2]$

Application to Physics

For a mass distribution with density $\rho(x)$ :

Total mass: $M = \int \rho(x)\,dx$
Center of mass: $\bar{x} = \frac{1}{M}\int x\rho(x)\,dx$
Moment of inertia: $I = \int x^2 \rho(x)\,dx$

Application to Signal Processing

For signals $f(t)$ and $g(t)$ :

Energy: $E = \int |f(t)|^2\,dt$
Cross-correlation: $(f \star g)(\tau) = \int f(t)g(t+\tau)\,dt$
Parseval's theorem connects time and frequency domains

Integral Properties Quiz

Questions

Correct

Accuracy

For

f, g \in R[a,b]

and

\alpha, \beta \in \mathbb{R}

\int_a^b [\alpha f + \beta g]\,dx

equals:

Easy

Not attempted

f(x) \leq g(x)

[a,b]

, then:

Easy

Not attempted

For

a < c < b

\int_a^b f\,dx

equals:

Easy

Not attempted

f \in R[a,b]

, is

|f| \in R[a,b]

Medium

Not attempted

The inequality

|\int_a^b f\,dx| \leq \int_a^b |f|\,dx

is called:

Easy

Not attempted

f, g \in R[a,b]

, is

fg \in R[a,b]

Medium

Not attempted

The Cauchy-Schwarz inequality states:

Hard

Not attempted

m \leq f(x) \leq M

[a,b]

, then:

Easy

Not attempted

Frequently Asked Questions

Why is linearity important?

Linearity allows us to break complex integrals into simpler parts and factor out constants, making computation much easier.

What's the geometric meaning of interval additivity?

The area under a curve from a to b equals the sum of areas from a to c and from c to b. This is intuitively obvious for positive functions.

Why does |f| integrable not imply f integrable?

Actually, the converse is more interesting: f integrable DOES imply |f| integrable. But knowing |f| is integrable doesn't tell us about f's integrability.

When is the triangle inequality an equality?

For |∫f| = ∫|f|, we need f to not change sign on [a,b]. If f changes sign, strict inequality holds.

How is Cauchy-Schwarz used in analysis?

It's used to bound products of integrals, prove convergence, and establish inequalities. It's the continuous analog of the vector dot product inequality.

Key Takeaways

Linearity

\int(\alpha f + \beta g) = \alpha\int f + \beta\int g

Additivity

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

Triangle Inequality

|\int f| \leq \int |f|

Estimation

m(b-a) \leq \int_a^b f \leq M(b-a)

Cauchy-Schwarz

(\int fg)^2 \leq \int f^2 \cdot \int g^2

Product Rule

$f, g \in R[a,b] \Rightarrow fg \in R[a,b]$

Decision Guide: Which Property to Use?

Breaking apart: Use linearity

Bounding: Use estimation theorem

Products: Use Cauchy-Schwarz

Symmetric interval: Use odd/even properties

Splitting domain: Use additivity

Absolute values: Use triangle inequality

18. Challenge Problems

Challenge 1: Proving Inequalities

Prove that for $f \in R[0,1]$ with $f \geq 0$ :

\left(\int_0^1 f(x)\,dx\right)^2 \leq \int_0^1 f(x)^2\,dx

Hint: Use Cauchy-Schwarz with $g(x) = 1$ .

Challenge 2: Limit Evaluation

Evaluate using integral properties:

\lim_{n \to \infty} \int_0^1 \frac{nx}{1 + n^2x^2}\,dx

Hint: Bound the integrand and use comparison.

Challenge 3: Hölder's Inequality

Prove the generalized Hölder inequality for $\frac{1}{p} + \frac{1}{q} = 1$ :

\int |fg| \leq \left(\int |f|^p\right)^{1/p} \left(\int |g|^q\right)^{1/q}

Note: Cauchy-Schwarz is the special case $p = q = 2$ .

Challenge 4: Integral Mean

Show that if $f$ is continuous and $\int_a^b f(x)g(x)\,dx = 0$ for all continuous $g$ , then $f \equiv 0$ .

Hint: Consider $g = f$ .

19. Connections to Higher Analysis

L^p Spaces

The properties we studied generalize to $L^p$ spaces where $\|f\|_p = (\int |f|^p)^{1/p}$ . Minkowski's inequality ensures these are genuine norms.

Inner Product Structure

The $L^2$ space has an inner product $\langle f, g \rangle = \int fg$ . Cauchy-Schwarz becomes the familiar $|\langle f,g\rangle| \leq \|f\|\|g\|$ .

Completeness

Unlike $R[a,b]$ , the $L^p$ spaces are complete—every Cauchy sequence converges. This is crucial for functional analysis.

Measure Theory

The Lebesgue integral extends these properties to a much larger class of functions, enabling powerful theorems like dominated convergence.

Quick Decision Guide

When to Use Linearity

Breaking a complex integrand into simpler parts
Pulling out constant factors
Combining multiple integrals

When to Use Comparison

Establishing bounds on integrals
Proving convergence of improper integrals
Showing an integral is positive/negative

21. Final Summary

The properties of definite integrals form the foundation for all computational and theoretical work with integrals. Mastering linearity, comparison, and the various inequalities will make integration much more manageable and intuitive.

Remember that these properties extend naturally to improper integrals, Lebesgue integrals, and multidimensional integrals. The patterns you learn here will serve you well throughout advanced mathematics, physics, and engineering courses.

Practice applying these properties systematically—they will become second nature with experience.

Properties of Definite Integrals

1. Linearity

2. Comparison Theorems

3. Interval Additivity

4. Absolute Value Integrability

5. Product and Special Inequalities

6. Important Examples

7. Additional Examples

8. Practice Problems

Problem Set A: Linearity and Comparison

Problem Set B: Absolute Value and Products

Problem Set C: Applications

9. Deeper Insights

Why These Properties Matter

Linearity

Comparison

Absolute Value

Cauchy-Schwarz

10. Historical Context

Augustin-Louis Cauchy (1789-1857)

Hermann Schwarz (1843-1921)

Hermann Minkowski (1864-1909)

11. Common Mistakes

❌ Wrong direction in triangle inequality

❌ Assuming ∣f∣|f|∣f∣ integrable implies fff integrable

❌ Forgetting (b−a)(b-a)(b−a) in estimation

❌ Misusing Cauchy-Schwarz

12. Summary of Key Properties

13. What's Next?

14. More Advanced Examples

15. Study Tips

Master the Basics First

Practice Estimation

Remember Direction of Inequalities

Use Symmetry

16. Quick Reference Card

Key Formulas

Symmetry Properties

Preservation

Comparison

Important Limits

17. Extended Applications

Application to Probability

Application to Physics

Application to Signal Processing

Frequently Asked Questions

Why is linearity important?

What's the geometric meaning of interval additivity?

Why does |f| integrable not imply f integrable?

When is the triangle inequality an equality?

How is Cauchy-Schwarz used in analysis?

Linearity

Additivity

Triangle Inequality

Estimation

Cauchy-Schwarz

Product Rule

Decision Guide: Which Property to Use?

18. Challenge Problems

Challenge 1: Proving Inequalities

Challenge 2: Limit Evaluation

Challenge 3: Hölder's Inequality

Challenge 4: Integral Mean

19. Connections to Higher Analysis

Lp Spaces

Inner Product Structure

Completeness

Measure Theory

Quick Decision Guide

When to Use Linearity

When to Use Comparison

21. Final Summary

❌ Assuming $|f|$ integrable implies $f$ integrable

❌ Forgetting $(b-a)$ in estimation

L^p Spaces