Section 1: Riemann Integral

Definition 1.1: Partition

A partition $\Delta$ of $[a, b]$ is a finite set of points:

a = x_0 < x_1 < x_2 < \cdots < x_n = b

The subintervals are $[x_{k-1}, x_k]$ with lengths $\Delta x_k = x_k - x_{k-1}$ .

Definition 1.2: Riemann Sum

For a partition $\Delta$ and sample points $\xi_k \in [x_{k-1}, x_k]$ , the Riemann sum is:

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

Definition 1.3: Riemann Integral

A function $f$ is Riemann integrable on $[a, b]$ if there exists a number $I$ such that:

\forall \varepsilon > 0, \exists \delta > 0: \text{if } \|\Delta\| < \delta, \text{ then } |S_\Delta(f, \xi) - I| < \varepsilon

We write $\int_a^b f(x)\,dx = I$ .

Theorem 1.1: Integrability of Continuous Functions

If $f$ is continuous on $[a, b]$ , then $f$ is Riemann integrable on $[a, b]$ .

Section 2: Properties of Definite Integrals

Theorem 2.1: Linearity

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

\int_a^b kf(x)\,dx = k\int_a^b f(x)\,dx

Theorem 2.2: Additivity

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

Theorem 2.3: Reversal

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

Example 2.1: Basic Computation

Evaluate: $\int_0^1 x^2\,dx$

Solution: Using FTC with $F(x) = x^3/3$ :

\int_0^1 x^2\,dx = \frac{x^3}{3}\bigg|_0^1 = \frac{1}{3} - 0 = \frac{1}{3}

Section 3: Fundamental Theorem of Calculus

Theorem 3.1: FTC Part 1

If $f$ is continuous on $[a, b]$ , then the function:

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

is differentiable on $(a, b)$ and:

F'(x) = f(x)

Proof of FTC Part 1:

For $h > 0$ :

\frac{F(x+h) - F(x)}{h} = \frac{1}{h}\int_x^{x+h} f(t)\,dt

By MVT for integrals, $\exists c \in [x, x+h]$ with:

\int_x^{x+h} f(t)\,dt = f(c) \cdot h

As $h \to 0$ , $c \to x$ , so:

F'(x) = \lim_{h \to 0} \frac{f(c)h}{h} = f(x)

∎

Theorem 3.2: FTC Part 2 (Newton-Leibniz Formula)

If $F$ is an antiderivative of $f$ on $[a, b]$ , then:

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

Proof of Newton-Leibniz Formula:

Let $G(x) = \int_a^x f(t)\,dt$ . By FTC Part 1, $G'(x) = f(x)$ .

Since $F'(x) = f(x)$ and $G'(x) = f(x)$ , we have $(F - G)'(x) = 0$ .

Therefore $F(x) - G(x) = C$ for some constant $C$ .

Since $G(a) = 0$ , we have $F(a) = C$ .

Thus $F(b) - G(b) = F(a)$ , so $G(b) = F(b) - F(a)$ . ∎

∎

Example 3.1: FTC Application

Evaluate: $\int_0^\pi \sin x\,dx$

Solution: Since $(-\cos x)' = \sin x$ :

\int_0^\pi \sin x\,dx = -\cos x\big|_0^\pi = -\cos \pi + \cos 0 = 1 + 1 = 2

Section 4: Mean Value Theorem for Integrals

Theorem 4.1: Mean Value Theorem for Integrals

If $f$ is continuous on $[a,b]$ , then there exists $c \in [a,b]$ such that:

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

Example 4.1: Application

For $f(x) = x^2$ on $[0,2]$ :

$\int_0^2 x^2\,dx = \frac{8}{3} = f(c) \cdot 2$

So $c^2 = \frac{4}{3}$ , giving $c = \frac{2\sqrt{3}}{3}$ .

Section 5: Integration by Substitution (Definite)

Theorem 5.1: Substitution Rule for Definite Integrals

If $u = g(x)$ and the derivative of $g$ is continuous on $[a,b]$ , then:

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

Example 5.1: Substitution

Evaluate: $\int_0^1 x(1+x^2)^3\,dx$

Let $u = 1+x^2$ , then $du = 2x\,dx$ :

\int_1^2 \frac{u^3}{2}\,du = \frac{1}{8}(16-1) = \frac{15}{8}

Section 6: Integration by Parts (Definite)

Theorem 6.1: Integration by Parts for Definite Integrals

\int_a^b u\,dv = uv\big|_a^b - \int_a^b v\,du

Example 6.1: Integration by Parts

Evaluate: $\int_0^\pi x\sin x\,dx$

Let $u = x$ , $dv = \sin x\,dx$ :

= -x\cos x\big|_0^\pi + \int_0^\pi \cos x\,dx = \pi + \sin x\big|_0^\pi = \pi

Section 7: Additional Properties

Theorem 7.1: Additivity

For any $c \in [a,b]$ :

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

Theorem 7.2: Comparison

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

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

Practice Quiz: Definite Integration

10

Questions

0

Correct

0%

Accuracy

1

What is a partition of

[a,b]

?

Easy

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2

The Riemann sum

S_\Delta(f, \xi)

equals:

Easy

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3

A bounded function

f

is Riemann integrable on

[a,b]

iff:

Medium

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4

The Newton-Leibniz formula states:

Easy

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5

If

f

is continuous at

x_0

, then

F'(x_0)

equals:

Easy

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6

\frac{d}{dx}\int_a^{x^2} \sin t\,dt

equals:

Medium

Not attempted

7

A continuous function on

[a,b]

is:

Easy

Not attempted

8

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

Easy

Not attempted

9

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

Easy

Not attempted

10

\int_0^1 x^2\,dx =

Easy

Not attempted

Frequently Asked Questions

What is the Riemann integral?

The Riemann integral is defined as the limit of Riemann sums as the partition becomes finer. A function f is Riemann integrable on [a,b] if the limit exists and is independent of the choice of sample points.

Why is FTC so important?

FTC connects the two main operations of calculus: differentiation and integration are inverse operations. This lets us evaluate definite integrals using antiderivatives instead of limits of Riemann sums.

What is the Newton-Leibniz formula?

If F is an antiderivative of f on [a,b], then ∫ₐᵇ f(x)dx = F(b) - F(a). This is the fundamental connection between definite and indefinite integrals.

What are the properties of definite integrals?

Key properties include: linearity (∫[f+g] = ∫f + ∫g), additivity (∫ₐᵇ + ∫ᵇᶜ = ∫ₐᶜ), reversal (∫ₐᵇ = -∫ᵇᵃ), and monotonicity (if f ≤ g, then ∫f ≤ ∫g).

Can the lower limit also vary?

Yes! Use ∫ₐˣ = -∫ˣₐ. For ∫_{g(x)}^{h(x)} f(t)dt, differentiate as: f(h(x))h'(x) - f(g(x))g'(x).