Concept of Calculus - Integral Calculus

• Integral Calculus is a key branch of mathematics.
• It deals with the properties of integrals and their applications.
• Physically, the integral of a function f(x) (i.e., ∫f(x)dx) represents the area under the curve of the function f(x) versus x.
• It is essentially the reverse process of differentiation.

The representation \[\int_{x=a}^{x=b}\] f(x)dx is called the definite integral of f(x) from x = a to x = b.

F(x) = ∫f(x) dx is called the indefinite (without any limits on x) integral of f(x).

• Reverse Process: Differentiation is the reverse process to that of integration.
• Area Under Curve: The physical interpretation of ∫f(x) dx is the area under the curve f(x) versus x.
• Definite Integral: It has specific limits on x (from x = a to x = b).
• Indefinite Integral: It does not have any limits on x.

When a simple formula (like for a rectangle or triangle) is not available:

Area under a straight line.

Area under a curve.

1. Divide the Area: The area under the curve is divided into a large number of vertical strips

2. Assume Rectangles: The thickness (width) of each strip is assumed to be so small that the strip can be treated as a rectangle.
3. Sum the Areas: The area under the curve is initially calculated as the sum of the areas of these rectangles:
   Area = \[\sum_{i=1}^n\Delta A_i=\sum_{i=1}^n(x_i-x_{i-1})f(x_i)\]
   where n is the number of strips and ΔA_i is the area of the i^th strip.
4. Take the Limit: Since the strips are only assumed to be rectangles, the sum is not exact. To get the exact area, the number of strips (n) is increased to infinity (n → ∞):
   Area = \[\lim_{n\to\infty}\sum_{i=1}^n(x_i-x_{i-1})f(x_i)\]
5. Use Integration: This limit process, which gives the exact area for a continuous change, is represented by the definite integral:
   \[\int_{x=a}^{x=b}f(x)dx=\lim_{n\to\infty}\sum_{i=1}^n(x_i-x_{i-1})f(x_i)\]

• Integration helps in getting the exact area under a curve, especially when the change is continuous (i.e., n is truly infinite).
• It is a crucial tool because it is the reverse operation of differentiation.

Fundamental Relation (Relating Differentiation and Integration)

• If F(x) = ∫f(x) dx, then:
  f(x) = \[\frac{d}{dx}(F(x))\]

Fundamental Theorem of Calculus (Definite Integral Evaluation)

• To evaluate a definite integral:
  \[F(x)|_a^b=F(b)-F(a)=\int_a^bf(x)dx\]

Properties of Integration

1. Sum Rule: \[\int(f_1(x)+f_2(x))dx=\int f_1(x)dx+\int f_2(x)dx\]
2. Constant Multiple Rule: \[\int Kf(x)dx=K\int f(x)dx\quad\mathrm{for~}K\] = Constant

Basic Indefinite Integral Formulas

Problem: Evaluate the following integrals:

1. ∫ x⁸ dx
2. \[\int_2^5x^2dx\]
3. ∫ (x + sin x) dx

Solution:

1. ∫ x⁸ dx
   Using the formula \[\int x^ndx=\frac{x^{n+1}}{n+1}{:}\]
   \[\int x^8dx=\frac{x^{8+1}}{8+1}=\frac{x^9}{9}\]
2. \[\int_2^5x^2dx\]
   Step 1: Find the indefinite integral using \[\int x^ndx=\frac{x^{n+1}}{n+1}{:}\]
   \[\int x^2dx=\frac{x^3}{3}\]
   Step 2: Apply the definite integral evaluation formula F(b)−F(a):
   \[\int_2^5x^2dx=\left.\frac{x^3}{3}\right|_2^5=\frac{5^3}{3}-\frac{2^3}{3}\]
   Step 3: Calculate the values:
   = \[\frac {125}{3}\] - \[\frac {8}{3}\] = \[\frac {117}{3}\]
   Step 4: Simplify the final result:
   = 39
3. ∫ (x + sinx) dx
   Step 1: Apply the Sum Rule (Eq. 2.45):
   \[\int(x+\sin x)dx=\int xdx+\int\sin xdx\]
   Step 2: Integrate each term using the basic formulas \[(\int x^{1}dx={\frac{x^{2}}{2}}\mathrm{and}\int\sin xdx=\] - cos x):
   = \[\frac {x^2}{2}\] + (-cos x)
   Step 3: Write the final expression:
   = \[\frac {x^2}{2}\] - cos x