Fundamental Theorem of Calculus

Fundamental Theorem of Calculus - Definitions

First Fundamental Theorem of Calculus

For 𝑓 a real-valued continuous function on an open interval 𝐼 and 𝑎 any number in 𝐼.

If 𝐹 is defined by:

• $F(x)={\int }_a^{x}f(t)dt$

then

• $F^{\prime }(x)=f(x)$

Second Fundamental Theorem of Calculus

If

• 𝑓 a real-valued continuous function on a closed interval [𝑎, 𝑏]
• 𝐹 is the indefinite integral of 𝑓 on [𝑎, 𝑏]

Then

• ${\int }_a^{b}f(x)dx=F(b)-F(a)$

Third Fundamental Theorem of Calculus

This applies to integrals along curves.

If 𝑓(𝑧) has a continuous indefinite integral 𝐹(𝑧) in a region 𝑅 containing a parameterized curve 𝛾: 𝑧 = 𝑧(𝑡) for 𝛼≤𝑡≤𝛽, then:

• ${\int }_{𝛾}f(z)dz=F(z(𝛽))-F(z(𝛼))$

Fundamental Theorem of Calculus - Proof

Click here to expand...

Interpreting the Behavior of Accumulation Function (TODO Why is this Here?)

Resources

• https://www.khanacademy.org/math/ap-calculus-ab/ab-integration-new/ab-6-4/v/fundamental-theorem-of-calculus