Calculus1-The Fundamental Theorem of Calculus, guaranteed 100% Pass

1


The Fundamental Theorem of Calculus


𝑥
Let 𝐴(𝑥) = ∫𝑎 𝑓(𝑡)𝑑𝑡 ; 𝑥 ≥ 𝑎 be the “net area” function for 𝑓(𝑡).



𝑦 = 𝑓(𝑡)




𝑎 𝑏 𝑐 𝑑




𝑏 𝑐 𝑑
𝐴(𝑏) = ∫𝑎 𝑓(𝑡)𝑑𝑡 , 𝐴(𝑐) = ∫𝑎 𝑓(𝑡)𝑑𝑡 , 𝐴(𝑑) = ∫𝑎 𝑓(𝑡)𝑑𝑡.


Ex. Suppose the graph of 𝑓(𝑡) is given below and 𝑎 = 0.
(1,4) (2,4)



𝑦 = 𝑓(𝑡)

(0,0)




(4, −4) (5, −4)

, 2


Find 𝐴(1), 𝐴(2), 𝐴(3), 𝐴(4), 𝐴(5).
1 1
𝐴(1) = ∫0 𝑓(𝑡)𝑑𝑡 = (1)(4) = 2
2
2
𝐴(2) = ∫0 𝑓(𝑡)𝑑𝑡 = 2 + 1(4) = 6
3 1
𝐴(3) = ∫0 𝑓(𝑡)𝑑𝑡 = 6 + (1)(4) = 8
2
4 1
𝐴(4) = ∫0 𝑓(𝑡)𝑑𝑡 = 8 − (1)(4) = 6
2
5
𝐴(5) = ∫0 𝑓(𝑡)𝑑𝑡 = 6 − 1(4) = 2.




Now we want to find 𝐴′(𝑥).
𝐴(𝑥+ℎ)−𝐴(𝑥)
𝐴′ (𝑥 ) = lim .
ℎ→0 ℎ

When ℎ is small:
𝐴(𝑥+ℎ)−𝐴(𝑥)
𝐴(𝑥 + ℎ) − 𝐴(𝑥) ≈ ℎ𝑓(𝑥) or ≈ 𝑓(𝑥) ;
ℎ

𝑦 = 𝑓(𝑡) 𝐴(𝑥 + ℎ) − 𝐴(𝑥)




𝐴(𝑥)




𝑎 𝑥 𝑥+ℎ