Fundamental Theorem of Calculus

Connect integration and differentiation. FTC Part 1 states that the derivative of the accumulation function is the rate function: $\frac{d}{dx} \int_a^x f(t)dt = f(x)$.

Interactive Sandbox

Rate vs Accumulator Graphs

Drag the upper boundary limit line **x** on either graph directly or adjust the slider. See that the accumulated shaded area (indigo) matches the height of the point on the lower graph, and the lower tangent slope (rose) equals the upper curve height.

Rate function: y = f(t) = t + 1
Accumulation function: F(x) = ∫ f(t) dt

FTC Part 1 Statement

d/dx [ ∫ax f(t) dt ] = f(x)
Upper Limit (x) 1.50
Drag the glowing vertical line on either canvas to dynamically update the integration bounds.
Algebraic Solver

Step-by-Step Solver

Evaluate integrals and verify the rate-slope relationship for the active bounds.

Integration & FTC Proof Steps


Accumulation Summary


Rate Value at boundary f(x)

f(1.50) = 2.50

Accumulated Area F(x)

F(1.50) = 2.625

Accumulator Tangent Slope F'(x)

F'(1.50) = 2.50

Evaluation

Theorem Quiz

Test your understanding of rate accumulations, FTC Part 1 derivative bounds, and integration values.

Question 1 of 5 Score: 0/0

According to FTC Part 1, what is the derivative d/dx of ∫₄x sin(t) dt?

Correct Answer!

Explanation text...

FTC Rules Summary


  • FTC 1: d/dx ∫ax f(t) dt = f(x)
  • FTC 2: ∫ab f(t) dt = F(b) - F(a)
  • FTC 1 Chain: d/dx ∫au(x) f(t) dt = f(u) u'
  • Accumulation F(x) = Shaded Area under rate curve