MATH1051 Lecture 5: Lecture 05
5.4 – Indefinite Integral and Net Change Theorem
Recall:
● – definite integral(x)x
∫
b
a
f
oThe result that we get from this definite integral is a number
▪From that number, we can get
●Net area
●Limit of Riemann Sums
●Net change
FTC (Fundamental Theorem of Calculus)
●If continuous on , and , then f a, ][ b(x) (t)tg =∫
x
a
f(x) (x)gI=f
● where (x)x
∫
b
a
f=F|b
aF′=f
Definition (Indefinite Integral)
●f(x)x(x)∫ = F
oMeans that if you take the derivative of , you’ll get (x)F(x)f
●Definite Integral
oYou’ll get a number
●Indefinite Integral
oWill result in a function
oDon’t forget the +C
● by FTC(x)dx
∫
b
a
f= ∫f(x)x|b
a
Examples
●xx∫2
o3
x3+C
oThis one is simple enough; take the integral and don’t forget + C
●θ∫ sin θ
cos θ
o⋅θ∫ sin θ
cos θ 1
sin θ
▪Split the fraction into two to begin using trig identities
ocot θ ⋅csc θ θ∫
Document Summary
5. 4 indefinite integral and net change theorem. Definite integral (x) x: the result that we get from this definite integral is a number. Ftc (fundamental theorem of calculus) g continuous on. , and f (x) b a f (x) x. = f (x: means that if you take the derivative of. , you"ll get f (x: you"ll get a number. Indefinite integral: will result in a function, don"t forget the + c (x)dx. = f(x) x|b a by ftc b a f. 2 o: this one is simple enough; take the integral and don"t forget + c x3 + c. Split the fraction into two to begin using trig identities cos . Sin o cot csc o o o o o ot sc cos = c sin . ) x x4 3 2 3 x | 1 sc : multiply the two then take the integral. Take the integral, then plug in the intervals to get the answer.
