What is the integral of an integral?

Question

Christopher Agresta · Accepted Answer

It really depends on what you mean by integral. Consider the indefinite integral of the indefinite intgeral of a function #f#: #int int f(x) dx dx# If #f(x) = g^(prime)(x) = h^(prime prime) (x)#, that is, #f# is a double antiderivative, then, by applying the Fundamental Theorem of Calculus twice: #int int f(x) dx dx = int int g^(prime) (x) dx dx = int [ g(x) + C ]dx = int [ h^(prime) (x) + C ]dx = int h^(prime) (x) dx + int C dx = h (x) + Cx + D,# where #C# and #D# are arbitrary constants. Now consider the definite integral of the definite integral of a function #f = g^(prime)(x)#. Then, applying the Fundamental Theorem of Calculus again: #int_c^d int_a^b f(x) dx dx = int_c^d int_a^b g^(prime) (x) dx dx = int_c^d [g(b)-g(a)] dx # But #g(b)-g(a)# is simply a real number.  For #int_c^d [g(b)-g(a)] dx# to make sense, we must consider #g(b)-g(a)# as a constant function.
Integrating again, we get: #int_c^d [g(b)-g(a)] dx = [g(b)-g(a)]x|_c^d = [g(b)-g(a)] (d-c)# A third interpretation could also be that of iterated integrals, that appear in multivariable calculus.
Consider the two variable function #f(x,y)#. There can be two kinds of iterated integrals: #int_(y_1)^(y_2) [ int_(x_1)^(x_2) f(x,y) dx ] dy# and #int_(x_1)^(x_2) [ int_(y_1)^(y_2) f(x,y) dy ] dx# If #f(x,y) = (partial)/(partial x) g_1(x,y)#, #g_1 (x_1,y)= d/(dy)h_1 (y)# and #g_1 (x_2,y)= d/(dy)p_1 (y)#, then we have the following result: #int_(y_1)^(y_2) [ int_(x_1)^(x_2) f(x,y) dx ] dy = int_(y_1)^(y_2) [(g_1(x_2,y)-g_1(x_1,y)] dy = [p_1(y_2)-p_1(y_1)] - [h_1(y_2) - h_1(y_1)]# An analogous result holds for #int_(x_1)^(x_2) [ int_(y_1)^(y_2) f(x,y) dy ] dx#, although, in general,  #int_(y_1)^(y_2) [ int_(x_1)^(x_2) f(x,y) dx ] dy ne int_(x_1)^(x_2) [ int_(y_1)^(y_2) f(x,y) dy ] dx#.

Christopher Agresta · Answer

undefined The integral of an integral, also known as a double integral, is a mathematical operation that calculates the area under a surface in a two-dimensional space. It involves integrating a function over a region in the plane. The result is a single value representing the volume between the surface and the xy-plane. Double integrals are commonly used in calculus, physics, and engineering to solve problems involving areas, volumes, and averages.