Applications of Integration

Antiderivatives

• The derivative of $F(x)$ is $f(x)$.

• The antiderivative of $f(x)$ is $F(x)$.

• $\int f(x) dx = F(x) + c$, where $\int$ is the integral sign, and $c$ is the constant of integration.

Rectilinear Motion

• Velocity function: $v(t) = \frac{d}{dt} s(t)$, where $s(t)$ is the position function.

• Acceleration function: $a(t) = \frac{d}{dt} v(t)$.

• Position function: $s(t) = \int v(t) dt$.

• Velocity function: $v(t) = \int a(t) dt$.

Definite Integrals

• Indefinite integral: $\int f(x) dx = F(x) + c$ (a function).

• Definite integral: $\int_{a}^{b} f(x) dx$ (a number).

• Evaluation: $\int_{a}^{b} f(x) dx = F(b) - F(a)$

Properties of Definite Integrals

• Switching limits: $\int{a}^{b} f(x) dx = -\int{b}^{a} f(x) dx$ .

• Equal limits: $\int_{a}^{a} f(x) dx = 0$ .

• Constant integral: $\int_{a}^{b} k dx = k(b - a)$, where k is a constant.

  • Example: $\int_{2}^{5} -8 dx = -8(5 - 2) = -24$

• Additive property: $\int{a}^{b} f(x) dx + \int{b}^{c} f(x) dx = \int_{a}^{c} f(x) dx$

Area Under a Curve

• Area = $\int_{a}^{b} f(x) dx$.

• Limit process: Area = $\lim_{n\to\infty}\sum{i=1}^{n}f(x_{i})\Delta x$

  • Riemann Sums: Approximating the area by breaking it up into rectangles.

  • $\Delta x$ = width of each rectangle.

  • $f(x_i)$ = height of each rectangle.

  • n = number of rectangles.

  • $\Delta x = \frac{b - a}{n}$

Summation Formulas

• $\sum_{i=1}^{n} c = cn$, where c is a constant.

• $\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$

• $\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$

• $\sum_{i=1}^{n} i^3 = (\frac{n(n+1)}{2})^2$

• $\sum_{i=1}^{n} i^4 = \frac{6n^5 + 15n^4 + 10n^3 - n}{30}$

• $\sum_{i=1}^{n} i^5 = \frac{2n^6 + 6n^5 + 5n^4 - n^2}{12}$

• Example: $\sum_{i=1}^{5} i^2 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 55$

  • Using the formula: $\frac{5(5+1)(2(5)+1)}{6} = \frac{5(6)(11)}{6} = 55$

Riemann Sums

• Left endpoints: $\Delta x [f(x0) + f(x1) + f(x2) + … + f(x{n-1})]$

• Right endpoints: $\Delta x [f(x1) + f(x2) + f(x3) + … + f(xn)]$

• Midpoint rule: $\Delta x [f(x{0.5}) + f(x{1.5}) + f(x{2.5}) + … + f(x{n-0.5})]$

• $\Delta x = \frac{b - a}{n}$

Approximate Integration

• Trapezoidal Rule: $Tn = \frac{\Delta x}{2} [f(x0) + 2f(x1) + 2f(x2) + … + 2f(x{n-1}) + f(xn)]$

• Simpson's Rule: $Sn = \frac{\Delta x}{3} [f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + … + 2f(x{n-2}) + 4f(x{n-1}) + f(xn)]$

• $\Delta x = \frac{b - a}{n}$

Fundamental Theorem of Calculus (FTC)

• Part 1: If $g(x) = \int_{a}^{x} f(t) dt$, then $g'(x) = f(x)$.

  • Explanation: $g(x) = F(x) - F(a)$, so $g'(x) = f(x) - 0 = f(x)$.

• Part 2: $\int_{a}^{b} f(x) dx = F(b) - F(a)$.

Net Change Theorem

• $\int_{a}^{b} F'(x) dx = F(b) - F(a)$

• Application: Accumulation of rate of change (e.g., water flow, velocity).

Area Between Two Curves

• In terms of x: Area = $\int_{a}^{b} [f(x) - g(x)] dx$, where $f(x)$ is the top function and $g(x)$ is the bottom function.

• In terms of y: Area = $\int_{c}^{d} [f(y) - g(y)] dy$, where $f(y)$ is the right function and $g(y)$ is the left function.

Volume by Disk Method

• Rotation about x-axis: $V = \pi \int_{a}^{b} [r(x)]^2 dx$, where $r(x)$ is the radius function.

• Rotation about y-axis: $V = \pi \int_{c}^{d} [r(y)]^2 dy$, where $r(y)$ is the radius function.