when we apply force on a spring as displacement increases value of force need to increase first when we push less displacement then less force needed as displacement of pushing goes on increasing needed more force to push spring you can experience here two quantities are involved one is displacement and other is force where displacement is independent and force depend upon displacement for more displacement more force needed so in mathematical language force is a function of x
f(x) where x is displacement here if asked what is work done then we can’t find work done by normal process that is work done = force *displacement = f*s why ? because here force is not constant as displacement is increases force value is increasing in every cm or mm displacement force value is increasing so which force value we will take answer is we don’t know hence in this situation if we want to calculate value of work done then our simple mathematics does’t support here integration support for calculation.
If i told you to calculate the area of this shape which formula you will use to calculate ? answer is there is no formula for this shape to calculate area we have only some standard shape formula for area calculation like triangle, circle, square, rectangle, sphere etc then how we will calculate this area, for this types of shape area calculation we have a very good method that is put this type of area on a graph paper see picture below.
here area of one square we know from graph paper and count total square and sum all square we will get the area of this irregular shape see here there are some square which is not covered by full square area for this we will take somewhere this square and left somewhere in counting the average will be same near to exact value but if you don’t accept this method then what we can do is make smallest square than previous square then it will be closet value its mean that whenever we make smallest and smallest square and sum of all the square in area our value will be accurate here some means integrate lets suppose one smallest square area is dA where dA→0 if we sum all this smallest areas we will get final area 𝗇
value so in mathematics we can write A = ∑dA
important Concept here for exact value we will make dA→0 means very very smallest value then value of n will be very very big number hence we can say n approaching to infinity n→ ∞ this time our result will be exact at this point either of square will be filled or vacant and this is only possible when dA→0
now at this point ∑ is written as Sum and S is written as ⨜ so we have added all square hence called sum and integration sign ⨜
now we know that this integration is apply where force is variable with respect to displacement see picture below.
suppose here f(x) is force along y axis and x is displacement along x axis very small displacement dx then its area will be A =work done = ⨜f(x)dx
Now integration is reverse process of differentiation how ?
differentiation means difference -division or integration means product sum so here division reverse product (multiplication) difference reverse sum so we can write differentiation = dy/dx or integration = ⨜ydx hence these two are reverse process of each other in physics we use both wherever application required .
⨜x^ndx = x^(n+1)/(n+1) remember this basic formula.
now there are two type of integration one is indefinite integral and other is definite integral when we write ⨜ydx = ⨜f(x)dx or
w = ⨜(3x-2)dx after solving we will get 3x²/2 -2x +c hence work done w = 3x²/2 -2x +c where c is constant of integration which is again a function of x no exact value of work done how much it depend upon value of x so it is indefinite integral because it may be any value for different value of x means no certain value but for definite integral its value is certain and limit is given between the range lower to upper limit so for definite integral we will get definite value like 10, 20, 100, 200..hence it is called definite integral thanks for reading .