Fundamental Theorem of Calculus II
Evaluate integrals using antiderivatives.
Key idea
Evaluate integrals using antiderivatives.
In depth
∫ₐᵇ f(x) dx = F(b) − F(a) for any antiderivative F. It turns the area problem into the antiderivative problem, avoiding Riemann sums.
Worked example
∫₁³ 2x dx = [x²]₁³ = 9 − 1 = 8.
Formula
Real-world uses
- Calculating exact areas, total change and averages in physics and economics
Common mistake
Computing F(a) − F(b) in the wrong order.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Fundamental Theorem of Calculus II?
Evaluate integrals using antiderivatives. It is a topic at the expert level in Calculus.
What is the formula for Fundamental Theorem of Calculus II?
The key formula is ∫ₐᵇ f = F(b) − F(a).
Where is Fundamental Theorem of Calculus II used in real life?
Real-world examples: Calculating exact areas, total change and averages in physics and economics.
What should I learn before Fundamental Theorem of Calculus II?
Make sure you are comfortable with Fundamental Theorem of Calculus I, the previous topic in Calculus.
What is a common mistake with Fundamental Theorem of Calculus II?
Computing F(a) − F(b) in the wrong order.
How can I practise Fundamental Theorem of Calculus II?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Fundamental Theorem of Calculus II?
Move on to Properties of Definite Integrals next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.