Lesson 15: Introduction to Integrals – Antiderivatives & Area

1. Antiderivatives – Reverse of Derivatives

If \(F'(x) = f(x)\), then \(F\) is an antiderivative of \(f\).

Notation: \(\int f(x) dx = F(x) + C\) (indefinite integral)

\(C\) is the constant of integration (family of curves).

Example:

\(\int 2x dx = x^2 + C\)
\(\int 3x^2 dx = x^3 + C\)

Exercise 1

2. Definite Integral – Area Under Curve

Definite integral from \(a\) to \(b\): \(\int_a^b f(x) dx = F(b) - F(a)\)

Fundamental Theorem of Calculus (Part 1):

If \9F\) is antiderivative of \(f\), then \(\int_a^b f(x) dx = F(b) - F(a)\)

Geometric meaning: net signed area between curve and \(x\)-axis from \(a\) to \(b\).

Exercise 2

Exercise 3

Which statements are true about integrals? (Select all that apply)

Antiderivative is reverse of derivative Definite integral gives net signed area \(\int f(x) dx\) always has \(+C\) Fundamental Theorem connects derivative and integral Integrals are only for positive functions

Exercise 4

Final challenge: match the application to the calculus concept (drag to first empty zone):

Gradient descent → Drop here
Velocity & acceleration → Drop here
Area under loss curve → Drop here
Tangent line approximation → Drop here

Click / tap any item to place it in the first empty zone:

Optimization / critical points

Derivatives (first & second)

Definite integral

Derivative (tangent slope)

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