Definite Integral

1. Basics of Definite Integral:

• Definition (NCERT Class 12, Chapter 7): Definite integral of a function ( f(x) ) with respect to (x) over an interval ([a, b]) is defined as the limit of the sum of areas of the rectangles approximating (f(x)) between (a) and (b) as the number of rectangles approaches infinity. $$ ∫a^b f(x) \ dx= \lim{n \to ∞} \sum_{i=1}^n f(c_i) \Delta x_i$$ where (c_i) is any point in the (i^{th}) subinterval with length (\Delta x_i), (n) represents the number of subintervals, and (c_i) is any intermediate value of (i^{th}) sub interval ([x_{i-1},x_{i}]).

• Properties:

• Linearity: (∫_a^b (f(x)+g(x)) \ dx=∫_a^bf(x) \ dx +∫_a^bg(x) \ dx).

• Additivity: (∫_a^b f(x) \ dx=∫_a^c f(x) \ dx +∫_c^b f(x) \ dx), for (a<c<b).

• Comparison Theorems: If (f(x) ≤ g(x)) for all (x) in ([a, b]), then (∫_a^b f(x) \ dx ≤∫_a^b g(x) \ dx). And, if ( f(x) ≥ g(x) ) for all (x) in ([a, b]), then ( ∫_a^b f(x) \ dx ≥ ∫_a^b g(x) \ dx ).

2. Techniques of Integration:

2.1 Integration by substitution:

• Rule: If (u = g(x)) is a differentiable function of (x) and (f) is a continuous function of (u), then $$ ∫f(g(x))g′(x) \ dx= ∫f(u) \ du.$$

• Examples:

(∫ sin^2 3x \ dx = - (1/6) cos 3x +C, let u = 3x) (∫ e^{5x} \ dx = (1/5)e^{5x}+C, let u = 5x)

2.2 Integration by parts:

• Rule: If (u) and ( v ) are differentiable functions of (x), then $$∫udv = uv - ∫vdu$$

• Examples:

(∫x \sin x dx = x (-\cos x) - ∫(-\cos x) dx= -xcos x +sinx + C) (∫ln x dx = xln x - ∫x \frac{1}{x} dx = x \ln x - x+C)

2.3 Trigonometric substitutions:

• Use the following substitutions to integrate trigonometric functions:

  • (sin θ = x , \ θ=arcsin x)

• (cos θ = x , \ θ=arccos x)

• (tan θ = x, \ θ = arctan x)

• Examples:

( ∫\sqrt{1-x^2 } dx = \frac{1}{2} (x \sqrt{1-x^2 } +\arcsin x )+ C ) ( ∫\frac{1}{x^2 + 1} dx = arctan x+ C)

2.4 Integration of rational functions:

• Use partial fraction decomposition to integrate rational functions.

2.5 Integration of irrational functions:

• Factor the integrand and use substitution to integrate functions with square roots or higher roots.

3. Applications of Definite Integral:

3.1 Area Under a Curve:

• For continuous and non-negative function (f(x)), the definite integral (\int_a^b f(x) \ dx) gives the area between the graph of (f(x)), the (x)-axis, and the vertical lines (x = a) and (x = b).

3.2 Volume of a Solid of Revolution:

• The volume generated by revolving the region between the graph of a positive continuous function (f(x)), the (x)-axis, and the vertical lines (x = a) and (x = b) about the (x)-axis is ( V = \pi ∫_a^b (f(x))^2 \ dx).

3.3 Length of a curve:

• For smooth curve with continuous derivatives such that the portion of the curve between (a) and (b) is given by (f(x)). The length of the curve over that portion is given by $$L=∫_a^b \sqrt{1+[f’(x)]^2 } dx$$

3.4 Work done by a force:

• The work done by a constant force (F) in moving an object through a distance (d) is $$W=F.d$$
• When a variable force (F(x)) is applied to an object and is displacing an object through a distance through the displacement (d) (i.e., from position (x_a) to position (x_b)), then the work ( W) done by the force is given by $$W = ∫_a^b F(x) \ dx $$

3.5 Average value of a function:

• The average value of a continuous function ( f(x) ) over the interval ([a, b]) is given by $$ f_{avg} =\frac{1}{b-a}∫_a^b f(x) \ dx.$$

4. Improper Integrals

4.1 Definition:

• Type 1: The improper integral (\int_a^∞ f(x) \ dx) is said to be convergent if ( \lim_{x \to \infty}∫_a^x f(x) \ dx ) exists as a finite number.

• Type 2: The improper integral (\int_a^b f(x) \ dx) is said to be convergent if ( \lim_{b \to \infty}∫_a^b f(x) \ dx ) exists as a finite number.

• Type 3: If the function has an infinite discontinuity at (c) in ((a,b)) and satisfies the convergence condition for either ( [a,c] ) or ( [c,b] ), then the improper integral ( \int_a^b f(x) \ dx ) is convergent.

4.2 Comparison test:

• The improper integral (\int_a^∞ f(x) \ dx ) is convergent if there exists a convergent improper integral (\int_a^∞ g(x) \ dx ) such that (0≤f(x)≤g(x)) for all (x≥a).

4.3 Applications

• Improper integrals are used to calculate volumes and areas of regions with infinite boundaries, convergence and divergence of infinite series.

5. Definite Integrals in 2D and 3D:

• Double Integral( over rectangular regions): For continuous function (f(x,y)) on a rectangular regions ( R= [a,b] \times [c,d] ), $$ ∬R f(x,y) \ dA =\lim{m\to ∞} \lim_{n\to ∞} \sum_{i=1}^m \sum_{j=1}^n f(c_i, d_j) \Delta x \Delta y$$ $$= \int_a^b \int_c^d f(x,y) \ dy \ dx$$

• Double Integrals over general regions:

• To evaluate a double integral over a region with curved or irregular boundaries, we use the following procedure:

  • Divide the region into small rectangular or triangular subregions.
  • Set up a definite integral for each subregion.
  • Add the definite integrals for each subregion.
  • Take the limit as the size of the subregions goes to zero.

• Triple Integral ( Rectangular Boxes): Similar to double integrals, triple integral over the rectangular box ( [a,b] \times [c,d] \times [e,f] ) $$\iiint_P f(x,y,z) \ dV = \int_a^b \int_c^d \int_e^f f(x,y,z) \ dz \ dy \ dx $$ Where (P) is a region in the three-dimensional space.

• Triple Integral over General Regions:

• Use the same divide integrate