Integral Calculus &Vector Calculus Rpp

Publication: Ramprasad Publication, Agra
Writers:

• Dr. Harikishan (K.R. {P.G.} College, Mathura)
• Dr. Vidya Sagar Chaubey (B.R.D. {P.G.} College, Deoria)
• Dr. Vijai Shankar Verma (D.D.U. Gorakhpur University, Gorakhpur)
• Dr. Prateek Mishra (M.L.K.{P.G.} College, Balrampur)

Unit V: Definite Integrals and Riemann Integration

1. Definite Integral as Limit of the Sum
   • Introduction to definite integrals.
   • Definite integral viewed as the limit of a sum.
2. Riemann Integral
   • Definition of the Riemann integral.
   • Integrability of continuous and monotonic functions.
   • Fundamental Theorem of Integral Calculus.
   • Mean value theorems related to integral calculus.
3. Differentiation Under the Sign of Integration
   • Differentiating a function that involves an integral.

Unit VI: Improper Integrals and Special Functions

4. Improper Integrals
   • Classification of improper integrals.
   • Tests for convergence: Comparison test, u-test, Abel’s test, Dirichlet’s test, Quotient test.
5. Beta and Gamma Functions
   • Definitions and properties of Beta and Gamma functions.
   • Application of Beta and Gamma functions in solving integrals.

Unit VII: Applications of Integration

6. Rectification
   • Calculating the length of curves (arc length).
7. Volumes and Surfaces of Solids of Revolution and Pappus’s Theorem
   • Determining volumes and surface areas of solids formed by revolving curves around an axis.
   • Use of Pappus’s theorem to find these values.
8. Multiple Integrals
   • Introduction to double and triple integrals.
   • Applications of Dirichlet’s Theorem and Liouville’s Theorem for multiple integrals.
9. Change of Order of Integration
   • Techniques to change the order of integration in double integrals.

Unit VIII: Vector Calculus

1. Vector Differentiation
   • Basic rules and operations of vector differentiation.
2. Gradient, Divergence, and Curl
   • Understanding the gradient of a scalar field, divergence, and curl of vector fields.
   • Calculating the normal on a surface.
   • Directional Derivative: The rate of change of a function in a specific direction.
3. Vector Integration
   • Basic operations in vector integration.
4. Theorems of Gauss, Green, and Stokes
   • Statements and applications of:
     • Gauss’s Theorem (also known as the Divergence Theorem).
     • Green’s Theorem (relating a line integral around a simple curve to a double integral).
     • Stokes’s Theorem (generalizing Green’s Theorem to higher dimensions).
   • Solving related problems using these theorems.

This book provides comprehensive coverage of Integral Calculus and Vector Calculus for undergraduate students. It introduces core topics such as definite and improper integrals, Beta and Gamma functions, and vector differentiation, building a solid foundation for the understanding of advanced calculus. The application of multiple integrals, as well as vector field theory through Gauss’s, Green’s, and Stokes’s theorems, equips students with powerful mathematical tools for both theoretical and practical problems.