Integral means that numbers are given to displacement, area, and volume. An integral notation is ∫ 2xdx where ∫ is the symbol of integral, 2x is the integration function and dx shows variable at the x-axis and dy at the y-axis. 

Firstly, to calculate integral there will be the division of the area, and all parts are added up with width Δx. To have an accurate answer the width of all parts must approach zero. 

In a definite integral, we have to know about 2 intervals that is a and b whose values will be specified. Its formula is following 

∫ (a, b) 2xdx

In indefinite integral, the values are not specified.  Its formula following 

∫ (n) 2xdx

Double integration works in a 3D and 2D space. It is having 2 variables with a region R. Its formula is following 

∫ ∫ R f (x, y) dxdy = lim n – 0 ∑ (n, i = 1) f (xi, yi) ∑xi ∑yi

One can find double integration of various types that are the following:

When we have to find the sum of the two functions

∬R [f (x, y) +g (x, y)] dA = ∬Rf (x, y) dA +∬Rg (x, y) dA

When we have to find out the difference between the two functions 

∬R [f (x, y) −g (x, y)] dA = ∬Rf (x, y) dA −∬Rg (x, y) dA

When there is the availability of factor that is constant 

∬Rkf (x, y) dA = k∬Rf (x, y) dA

When a solid is given and we have to find its volume

If f (x, y) ≥ g (x, y) 

If its region is given it would be as

V=∬R [f (x, y) −g (x, y)] dA

Derivation means that when an argument changes, function values change itself. Its formula is following 

f′(x0) = limΔx → 0Δy / Δx = limΔx → 0; f(x0+Δx) −f(x0) / Δx

Its types are Lagrange’s notation, Leibniz’s notation. In the first one notation, if we have to the f derivative it is going to be as Y function. As following Y = f(x) as f′(x) or y′(x). 

In the second notation, the function Y becomes

Y = f(x) as df / dx or dy / dx.

There are six rules of derivation.

First, is the constant rule which is following 

f(x) = C   then f ′(x) which becomes equal to 0

Second, is the multiple constant rule which is following 

g(x) = C * f(x) then g′(x) = c · f ′(x)

Third, is the difference rule which is following 

h(x) = f(x)±g(x) then h′(x) = f ′(x) ± g′(x)

Fourth, is the product rule which is following 

h(x) = f(x)g(x) then h′(x) = f ′(x) g(x) + f(x) g′(x)

Fifth, is the quotient rule which is following 

h(x) = f(x)/g(x) then, h′(x) = f ′(x) g(x) − f(x) g′(x) / g(x)²

Sixth, is the chain rule which is following 

h(x) = f(g(x)) then h′(x) = f ′ (g(x)) g′(x)

We can find the derivations of the following 

Exponential, Sin, Cos, and Tan.

Exponential derivation:

We can find exponential derivation by following formulas:

f(x) = a˟  then; f ′(x) = ln(a)  a˟

f(x) = e˟ then; f ′(x) = e˟

f(x) = aᶢ˟ then    f ′(x) = ln(a)aᶢ˟ g′˟

f(x) = eᶢ˟ then   f ′(x) = eᶢ˟ g′(x)

Sin derivation:

We can find the derivation of Sin by the following formula

d / dx [Sin (x)] = Cos (x)

Cos derivation:

We can find the derivation of Cos by the following formula

d / dx [Cos (x)] = – Sin (x) 

Tan derivation:

We can find the derivation of Tan by the following formula 

Tan (x) = sec2x.

There are online calculators like integration calculator and derivative calculator which you can find online. All the available online calculators use the same formulas to give results. Online calculators can increase your efficiency and help you to save a lot of time.