Definite and Indefinite Integrals, Area Between Curves, and Substitution Techniques

Definite and Indefinite Integrals

Definite Integrals and the Fundamental Theorem of Calculus

The definite integral of a function over an interval represents the signed area under the curve between two points. The Fundamental Theorem of Calculus (Part 2) connects differentiation and integration, allowing us to evaluate definite integrals using antiderivatives.

• Definite Integral: $\int_a^b f(x)\,dx$ gives the area under $f(x)$ from $x=a$ to $x=b$.

• Fundamental Theorem of Calculus (Part 2): $\int_a^b f(x)\,dx = F(b) - F(a)$, where $F(x)$ is any antiderivative of $f(x)$.

• Steps:

  1. Find $F(x)$, the antiderivative of $f(x)$.

  2. Evaluate $F(b)$ and $F(a)$, then compute $F(b) - F(a)$.

Indefinite Integrals

An indefinite integral represents a family of functions whose derivative is the integrand. It is called the general antiderivative and includes a constant of integration.

• Indefinite Integral: $\int f(x)\,dx = F(x) + C$

• Purpose: To find a function whose derivative is $f(x)$.

Applications of Integration: Motion and Area

Position, Velocity, and Integration

Velocity is the first derivative of position with respect to time, and position is the antiderivative (integral) of velocity. Integration can be used to find position from velocity, given initial conditions.

• Given: $v(t)$ (velocity function), $s'(t) = v(t)$

• Find: $s(t) = \int v(t)\,dt$

• Initial Condition: Use $s(0)$ or another known value to solve for $C$.

• Example: For $v(t) = -(t-2)^2 + 1$, $s(t) = \int (-(t-2)^2 + 1)\,dt = -\frac{t^3}{3} + 2t^2 - 3t + C$

Interpreting Velocity and Position Graphs

• Positive Velocity: $v(t) > 0$ (particle moves right/up)

• Negative Velocity: $v(t) < 0$ (particle moves left/down)

• Turning Points: Where $v(t) = 0$ (change in direction)

• Leftmost/Rightmost Position: Minimum/maximum of $s(t)$ on the interval

Area Between Curves

General Formula

The area between two curves $y = f(x)$ and $y = g(x)$ from $x = a$ to $x = b$ is given by:

• $A = \int_a^b [f(x) - g(x)]\,dx$, where $f(x) \geq g(x)$ on $[a, b]$

• If $a$ and $b$ are not given, find intersection points by solving $f(x) = g(x)$.

• For regions where the top and bottom functions change, split the integral at those points.

Example:

• Find area between $y = 2x^2 - 1$ and $y = 1$ from $x = -1$ to $x = 1$:

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

Applications: Population, Airflow, and Average Value

Population Growth and Net Change

• Given birth and death rates $b(t)$ and $d(t)$, the net population change over $[0, T]$ is:

• $A = \int_0^T [b(t) - d(t)]\,dt$

• Example: $b(t) = 2200e^{0.024t}$, $d(t) = 1460e^{0.018t}$, $A = \int_0^{10} [2200e^{0.024t} - 1460e^{0.018t}]\,dt$

Average Value of a Function

• The average value of $f(x)$ on $[a, b]$ is:

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

• Example: Average temperature over a year using $T(t)$ and integrating over the interval.

Integration Techniques: Substitution

Substitution Method

Substitution is used to simplify integrals by changing variables, making the integral easier to evaluate.

• If $u = g(x)$, then $du = g'(x)dx$ and $\int f(g(x))g'(x)dx = \int f(u)du$

• For definite integrals, change the limits to $u$-values.

• For indefinite integrals, substitute back to $x$ after integrating.

Examples:

• $\int (x+1)^{1000} dx$ Let $u = x+1$, $du = dx$ $\int u^{1000} du = \frac{u^{1001}}{1001} + C = \frac{(x+1)^{1001}}{1001} + C$

• $\int \sqrt{x^4+1} 4x^3 dx$ Let $u = x^4+1$, $du = 4x^3 dx$ $\int \sqrt{u} du = \int u^{1/2} du = \frac{2}{3} u^{3/2} + C = \frac{2}{3} (x^4+1)^{3/2} + C$

• Definite integral: $\int_1^3 5\sin(1-2x) dx$ Let $u = 1-2x$, $du = -2dx$, $-\frac{du}{2} = dx$ Change limits: when $x=1$, $u=-1$; when $x=3$, $u=-5$ $\int_1^3 5\sin(1-2x) dx = -\frac{5}{2} \int_{-1}^{-5} \sin(u) du = \frac{5}{2} [\cos(-5) - \cos(-1)]$

Summary Table: Key Integration Concepts

Concept	Formula	Notes
Definite Integral	$\int_a^b f(x)dx = F(b) - F(a)$	Area under curve, use antiderivative
Indefinite Integral	$\int f(x)dx = F(x) + C$	General antiderivative, includes constant
Area Between Curves	$\int_a^b [f(x) - g(x)]dx$	$f(x)$ above $g(x)$ on $[a, b]$
Average Value	$\frac{1}{b-a} \int_a^b f(x)dx$	Mean value of function on interval
Substitution	$u = g(x), du = g'(x)dx$	Change of variables to simplify integral

Additional info: These notes cover key integration concepts from Calculus, including definite and indefinite integrals, area between curves, applications to motion and population, average value, and substitution techniques. Examples and step-by-step solutions are provided for clarity.