Surface and Volume Integrals

To understand the concept of surface and volume integral we firstly need to know about integral calculus. In calculus, an integral is often utilised for calculating a function’s antiderivative. Hence a function’s antiderivative is often referred to as the function’s integral. The process through which the anti-derivative of a particular function is calculated is known as integration. In simple words, the computation of integrals is the inverse process of finding the derivatives of a particular function. A function’s integral shows or depicts a family of curves. Hence in this context, it should be mentioned that finding the computation of the integrals as well as the derivatives forms the basis of calculus.

Surface and Volume integral

Surface and volume integrals are vital concepts of integrals in mathematics, more specifically in multivariate calculus. Firstly a surface integral is defined in vector calculus as the generalisation of a wide range of integrals for integration over a surface. Mostly Surface integrals are viewed as double integrals. For a specifically provided surface, the integral can be calculated either for the vector field or the scalar field. If the scalar integral is calculated for a scalar field then the resulting value is scalar in nature. On the other hand, if the surface integral is calculated in a vector field then the resultant value is a vector in nature. Next, the concept of volume integral is a bit different from that of surface integrals. In multivariate calculus, volume integral is defined as the integral of a specific function in a domain that is three-dimensional. Hence, it can be considered a special as well as important case within multiple integrals. Through this, the volume of a particular region in 3-dimension can be calculated. Volume integral is particularly useful in Physics or more specifically in Agricultural engineering for calculating flux densities.

Integration of some special trigonometric function functions

This part will particularly discuss the different processes of integration through which some special trigonometric functions can be successfully integrated. The different special forms and the methods through which they can be solved have been outlined in the following.

• If the value of n is positive as well as odd then to evaluate

1. ∫ sinⁿx dx, we consider cos x as z.
2. ∫ cosⁿx dx, we consider sin x as z.

• If the value of n is positive as well as even integer, then for evaluating ∫ cosⁿx dx of ∫ sinⁿx dx, we must express sinⁿx or cosⁿx in the form of some multiple angle function.
• For evaluating ∫ sinⁿx cosⁿx dx, where m and n are both positive,

1. We put Cos x or sin x = y if both n and m are odd.
2. We put sin x = y if n is odd but m is even
3. We put cos x = y if m is odd but n is even
4. We must express sinⁿx cosⁿx as a multiple angle function, if both n, m are even
5. If n and m are real n + m as a negative integer that is even in nature then the integration can be done through substitution of tan x = y.

Integration of some special rational functions

This part will particularly discuss the different process of integration through which some special rational algebraic functions can be successfully integrated. The different special forms and their solved outcomes have been outlined in the following

• ∫ dy / (y² + b²) = (1/b) tan^-1 (y/b) + c ; ∫ dy / (y² + 1) =  tan^-1 y + c
• ∫ dy / (y² – b²) = (1/2b) log |(y – b) / (y + b)| + c, [|x| > |a|]
• ∫ dy / ( b² – y²) = (1/2b) log |(b + y) / (b – y)| + c, [|x| < |a|]
• ∫ dy / √(y² ± b²) = log |(y + √( y² ± b²)| + c
• ∫ dy / √( b² – y²) = sin-1 (y/b) + c; ∫ dy / √( 1 – y2) = sin-1 y + c

In the above formulas ‘b’ is any number and ‘c’ is constant.

Conclusion

Throughout the article, the core integral concept of surface and volume integral has been mostly discussed. These are important topics that utilise the concepts of multiple integrals. The topic has been further analysed through discussion of some special integrals as well as integration of some special forms.