Fundamental Theorem of Calculus (FTC)
Last updated
Last updated
This is somehow dreaded and mind-blowing. But it's the only thing to relate the
Differential Calculus&Integral Calculus.
It's so much clearer if you see the function in the middle of integration as a derivative.
Notice that: In this theorem, the lower boundary a is completely "ignored", and the unknown t directly changed to x.
1st FTC & 2nd FTCThe Fundamental Theorem of Calculus could actually be used in two forms. They have different use for different situations.
(Notice that boundaries & terms are different)
How to Differentiate IntegralsWe could CONVERT the integral formula to Differential formula, by using the fundamental theorem of calculus, and use the Rules we've learnt to solve the differential equations.
We got different strategies for different boundaries situation:
A variable and a number.
A function and a number.
Two functions.
▼ Here is formulas for different boundaries of integration:
It's to apply the boundary situation strategy of A variable & a number: G'(x) = g(x)
Assume the function in the middle of integral is G'(x) = 3x²+4x
Since it's asking for g'(x), so it's differentiate the Integral: d/dx ʃ G'(x) dt expression
So g'(2) = G'(x) = 3x²+4x = 20
According to the different Boundary situation strategies, here we apply the A function & a number strategy: F'(x) = f[g(x)] · g'(x)
So F'(x) = √(15 - 2x) · (2x)' = 2√(15-2x)
It's asking you to apply the FTC in form of d/dx ʃ f'(x) dx = f(b) - f(a)
So it becomes calculating F(3) - F(0) = 125 - 1 = 124
We could use the Second Fundamental Theorem of Calculus:
which in this case is:
And we move the known terms to one side and keep the asking term at another side:
Solve:
Solve:
Solve:
Solve:
