Integral Calculus

Example on Properties of Definite Integration

• Property 1: Linearity of Definite Integration

  • If $f(x)$ and $g(x)$ are integrable functions and $c$ is a constant, then
  • $\int_{a}^{b} cf(x) + g(x) : dx = c \int_{a}^{b} f(x) : dx + \int_{a}^{b} g(x) : dx$

• Property 2: Change of Limits

  • If $f(x)$ is integrable on $[a, b]$ and $c$ and $d$ are constants, then
  • $\int_{a}^{a} f(x) : dx = 0$
  • $\int_{a}^{b} f(x) : dx = -\int_{b}^{a} f(x) : dx$
  • $\int_{a}^{b} f(x) : dx = \int_{a}^{c} f(x) : dx + \int_{c}^{b} f(x) : dx$

• Property 3: Definite Integral of a Constant

  • If $f(x) = c$, a constant, then
  • $\int_{a}^{b} c : dx = c(b - a)$

• Property 4: Additivity of Definite Integration

  • If $f(x)$ is integrable on $[a, b]$ and $c$ is a constant, then
  • $\int_{a}^{b} f(x) : dx = \int_{a}^{c} f(x) : dx + \int_{c}^{b} f(x) : dx$

• Property 5: Conservation of Definite Integrals under Limits Transformation

  • If $f(x)$ is integrable on $[a, b]$ and $c$ and $d$ are such that $f(c + d - x)$ is integrable on $[a, b]$, then
  • $\int_{a}^{b} f(x) : dx = \int_{a}^{b} f(c + d - x) : dx$

11. Property 6: Definite Integral of the Sum or Difference of Functions
    • If $f(x)$ and $g(x)$ are integrable on $[a, b]$, then
    • $\int_{a}^{b} [f(x) \pm g(x)] : dx = \int_{a}^{b} f(x) : dx \pm \int_{a}^{b} g(x) : dx$

12. Property 7: Definite Integral of the Product of a Function and a Constant
    • If $f(x)$ is integrable on $[a, b]$ and $c$ is a constant, then
    • $\int_{a}^{b} [c \cdot f(x)] : dx = c \int_{a}^{b} f(x) : dx$

13. Property 8: Definite Integral of the Product of Two Functions
    • If $f(x)$ and $g(x)$ are integrable on $[a, b]$, then
    • $\int_{a}^{b} [f(x) \cdot g(x)] : dx \neq \int_{a}^{b} f(x) : dx \cdot \int_{a}^{b} g(x) : dx$

14. Property 9: Change of Variable in Definite Integral
    • If $f(x)$ is integrable on $[a, b]$ and $x = \varphi(t)$ is a differentiable and monotonic function mapping $[c, d]$ onto $[a, b]$, then
    • $\int_{a}^{b} f(x) : dx = \int_{c}^{d} f(\varphi(t)) \cdot \varphi’(t) : dt$

15. Property 10: Definite Integral of a Non-Negative Function
    • If $f(x)$ is integrable on $[a, b]$ and $f(x) \geq 0$ for all $x$ in $[a, b]$, then
    • $\int_{a}^{b} f(x) : dx \geq 0$

16. Property 11: Definite Integral of an Even Function
    • If $f(x)$ is integrable on $[-a, a]$ and $f(x) = f(-x)$ for all $x$ in $[-a, a]$, then
    • $\int_{-a}^{a} f(x) : dx = 2\int_{0}^{a} f(x) : dx$

17. Property 12: Definite Integral of an Odd Function
    • If $f(x)$ is integrable on $[-a, a]$ and $f(-x) = -f(x)$ for all $x$ in $[-a, a]$, then
    • $\int_{-a}^{a} f(x) : dx = 0$

18. Trigonometric Integral Examples
    • Example 1: $\int_{0}^{\pi} \sin^2(x) : dx$
    • Example 2: $\int_{0}^{\pi} \cos^2(x) : dx$
    • Example 3: $\int_{0}^{\pi} \sin^3(x) : dx$
    • Example 4: $\int_{0}^{\pi} \cos^4(x) : dx$

19. Exponential Integral Examples
    • Example 1: $\int_{0}^{1} e^x : dx$
    • Example 2: $\int_{0}^{2} e^{2x} : dx$
    • Example 3: $\int_{0}^{1} xe^{-x^2} : dx$
    • Example 4: $\int_{0}^{\ln 2} e^{2x} : dx$

20. Application: Area Under a Curve
    • The area $A$ under the curve $y = f(x)$ between $x = a$ and $x = b$ can be calculated using a definite integral.
    • $A = \int_{a}^{b} f(x) : dx$
    • Example: Find the area under the curve $y = x^2$ between $x = 0$ and $x = 1$.

21. Application: Average Value of a Function
    • The average (mean) value of a function $f(x)$ on the interval $[a, b]$ is given by
    • $f_{\text{avg}} = \frac{1}{b - a} \int_{a}^{b} f(x) : dx$
    • Example: Find the average value of the function $f(x) = 3x^2 - 4x + 2$ on the interval $[1, 4]$.

22. Application: Length of a Curved Line
    • The length of a curve $y = f(x)$ on the interval $[a, b]$ can be calculated using the formula
    • $L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} : dx$
    • Example: Find the length of the curve $y = x^{\frac{3}{2}}$ on the interval $[0, 4]$.

23. Application: Volume of Revolution
    • The volume of the solid generated by revolving the area bounded by the curve $y = f(x)$ and the x-axis from $x = a$ to $x = b$ about the x-axis is given by
    • $V = \pi \int_{a}^{b} [f(x)]^2 : dx$
    • Example: Find the volume of revolution generated by revolving the curve $y = \sqrt{x}$ from $x = 0$ to $x = 4$ about the x-axis.

24. Application: Work Done
    • The work done in moving an object along a curve $y = f(x)$ against a force field given by $F(x)$ on the interval $[a, b]$ is given by
    • $W = \int_{a}^{b} F(x) \cdot dx$
    • Example: Find the work done in moving an object along the curve $y = x^2$ from $x = 0$ to $x = 2$ against the force field $F(x) = 3x$.

25. Application: Moment of Inertia
    • The moment of inertia of a thin rod of length $L$ and mass per unit length $\rho(x)$ about an axis passing through one end and perpendicular to the rod is given by
    • $I = \int_{0}^{L} [x \cdot \rho(x)]^2 : dx$
    • Example: Find the moment of inertia of a thin rod of length $L$ and mass per unit length $\rho(x) = kx$ about an axis passing through one end and perpendicular to the rod.

26. Application: Probability Density Function
    • For a continuous random variable $X$ with a probability density function $f(X)$ defined on the interval $[a, b]$, the probability of $X$ lying within the interval $[c, d]$ is given by
    • $P(c \leq X \leq d) = \int_{c}^{d} f(X) : dX$
    • Example: Find the probability of a continuous random variable $X$ lying within the interval $[1, 3]$ if the probability density function is given by $f(X) = \frac{1}{4}$.

27. Application: Population Growth
    • The change in population $P(t)$ over time can be modeled using the equation
    • $\frac{dP}{dt} = k \cdot P(t)$
    • The solution to this equation is given by
    • $P(t) = P_0 \cdot e^{kt}$
    • Example: Solve the population growth problem for a population with initial population $P_0 = 100$ and growth rate $k = 0.05$.

28. Application: Heat Transfer
    • The rate of change of heat $Q(t)$ with respect to time can be modeled using the equation
    • $\frac{dQ}{dt} = k \cdot A \cdot \Delta T$
    • The solution to this equation is given by
    • $Q(t) = Q_0 + k \cdot A \cdot (T - T_0)$
    • Example: Solve the heat transfer problem for an object with initial heat $Q_0 = 500$ and undergoing heat transfer at a rate $k = 0.1$ with surface area $A = 2$ and temperature difference $\Delta T = 50$.

29. Summary: Properties of Definite Integration
    • Linearity of Definite Integration
    • Change of Limits
    • Definite Integral of a Constant
    • Additivity of Definite Integration
    • Conservation of Definite Integrals under Limits Transformation
    • Definite Integral of the Sum or Difference of Functions
    • Definite Integral of the Product of a Function and a Constant
    • Definite Integral of the Product of Two Functions
    • Change of Variable in Definite Integral
    • Definite Integral of a Non-Negative Function

30. Summary & Review
    • Average Value of a Function
    • Length of a Curved Line
    • Volume of Revolution
    • Work Done
    • Moment of Inertia
    • Probability Density Function
    • Population Growth
    • Heat Transfer
    • Importance of Definite Integration in Various Applications
    • Practice Problems and Exercises