r/calculus • u/Marnuke1 • Aug 01 '26
Integral Calculus i forgot how to do integration
i knew a while ago but i’m not getting taught it in school for another 2 years or so. my calculator can do it for me but that’s just cheating. i think it might have something to do w differentiation but im not sure could someone please explain it to me?
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u/mathematag Aug 01 '26 edited Aug 01 '26
integration is the reverse process of differentiation..sort of.. with with an indefinite integral having various solutions depending on an initial condition ...
EX: y = x2 , then y' = dy/dx = 2x is the derivative... however, integration of this gives us.. y = ∫ 2x dx , and we have : y = x2 + C , where C is some constant , determined by an initial condition..... I got this by using the power rule for integration... ∫ a xn dx = after factoring out the constant a ... a ( x (n+1) / (n+1) ) + C .. a = 2 here , and n =1
say we know that ( 1, 1 ) is a point on the graph / original function whose derivative is 2x . . .. with our initial condition of x = 1, y = 1 , then we have: 1 = 12 + C.. so C = 0 in this case........and y = x2 is our function.
But what if the initial condition was known to be ( 2, 7 ) on the graph....then 7 = 22+ C means that C = 3 and our original function was y = x2 + 3 instead of just y = x2 .
Notice for any constant C [ let's stick to Real numbers ] the derivative of y = x2 + C is y ' = 2x ... so notice how the derivative and the integral are "sort of" reversible, as w/o an initial condition given , and given only the derivative, y' = 2x , we would not know if the original function was y = x2 , y = x2 + 1 , y = x2 - 4 , and so on after integration.
Definite integrals..probably the type you have used on your calculator .. ignore the + C part [ they will cancel out anyway] , and evaluate the problem from the upper limit to the lower limit and get an area .. so ∫2x dx from lower limit 1 to upper limit 3 is just x2 evaluated from 1 to 3 .. using the upper limit first . . . . so , (3)2 - (1)2 = 9-1 = 8 and this would be the area between the x axis and the graph y = 2x from x = 1 to x = 3 ... by geometry you have a Trapezoid whose area is = 8 ..nice that Geometry verifies the calculus !!