Question

how is area under curve formula related with differentiation

Answer 1

\[f zdx = 1/2fzdx + 1/2b y(b) - 1/2a y(a)\]

Answer 2

how is this related with differentiation

Answer 3

i think it will be easier to understand if u post the complete question

Answer 4

area under a curve is mainly formulated with integration

Answer 5

yeah but does it has any relation with differentiation

Answer 6

we were asked to show one relation

Answer 7

any relation

Answer 8

@sidsiddhartha

Answer 9

did u get anything

Answer 10

@sidsiddhartha

Answer 11

differentiation is mainly used to calculate the slope and integration to calculate area but u can use a little bit of limits like
consider a function f(b) continuous in (0,b)
and a master function
\[m(b)=\int\limits_{0}^{b}f(x)dx\]
so now i can use a little bit diffrentiation like--
\[\frac{ d }{ db }[m(b)]=\lim_{h \rightarrow 0}\frac{ m(b+h)-m(b) }{ h }=\lim_{h \rightarrow 0}\frac{ 1 }{ h }\int\limits_{b}^{b+h}f(x)dx=f(b)\]

Answer 12

thanks so much

Answer 13

did u calculate

Answer 14

no problem,but remember its valid for only a continuous function i think :)

Answer 15

did u calculate

Answer 16

which function

Answer 17

I understand how derivatives work based on the definition, and the fact that my professor explained it step by step until the point where I can derive it myself.

However when it comes to the area under a curve for some reason when you break it up into an infinite amount of rectangles, magically it turns into the anti-derivative. Can someone explain why that is the definition of the integral and how Newton figured this out?