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C23 · Unit C6 Integration and accumulation

Fundamental Theorem: differentiating accumulation

CoreReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 6.4–6.5
Builds on
Part of

Progress on this lesson counts in every pathway listed.

Understand the idea

If f is continuous, A(x) equals ∫ₐˣf(t)dt satisfies A minutes (x) equals f(x). The derivative of an accumulation function returns the current rate. For a moving upper bound g(x), chain rule gives d over dx∫ₐ to the power g (x)f(t)dt equals f(g(x))g minutes (x). A moving lower bound contributes a minus sign. A minutes and A seconds can analyze accumulation without evaluating an antiderivative.

Worked example

For A(x) equals ∫ sub 1 ^(x squared)cos t dt, A minutes (x) equals 2x cos(x squared). The lower bound contributes no derivative because it is constant. If B(x) equals ∫ₓ cubed e^(t squared)dt, then B minutes (x) equals minus e^(x squared), even though no elementary antiderivative is available.

Explore the mathematics

Fundamental Theorem: differentiating accumulation

Connect an accumulating area with its instantaneous rate.

Setting 3 of 5

Scroll sideways to see the full diagram.

Predict the rate at which accumulated area grows.xy123F(t)=∫₀ᵗ x² dx; t=2

Predict before continuing

Predict the rate at which accumulated area grows.

Read the complete explanation
  1. Predict the rate at which accumulated area grows.
  2. A small increase Δt adds a thin strip with approximate area t²Δt.
  3. Dividing by Δt and taking the limit gives F′(t)=t².
  4. At t=2, F′(t)=4. For a moving upper limit g(t), multiply by g′(t).
Fundamental Theorem: differentiating accumulation · visual worked method
Fundamental Theorem: differentiating accumulation · visual worked methodFor A(x)=∫₁^(x²)cos t dt, A′(x)=2x cos(x²). The lower bound contributes no derivative because it is constant. If B(x)=∫ₓ³e^(t²)dt, then B′(x)=−e^(x²), even though no elementary antiderivative is available.Worked example · 1/2For A(x)=∫₁^(x²)cos t dt, A′(x)=2xcos(x²). The lower boundcontributes no derivative becauseit is constant. IfWorked example · 2/2B(x)=∫ₓ³e^(t²)dt, thenB′(x)=−e^(x²), even though noelementary antiderivative isavailable.

For A(x)=∫₁^(x²)cos t dt, A′(x)=2x cos(x²). The lower bound contributes no derivative because it is constant. If B(x)=∫ₓ³e^(t²)dt, then B′(x)=−e^(x²), even though no elementary antiderivative is available.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

The integration variable t is a dummy variable; do not substitute x into the bounds and integrand indiscriminately.

Investigate and explain

Given a signed graph of f, predict where its accumulation function increases, decreases and changes concavity before finding any formulas.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.