Unit 6 Notes: Understanding the Fundamental Theorem of Calculus (AP Calculus BC)

Accumulation function

A function defined by a definite integral with a variable limit (e.g., g(x)=∫_a^x f(t)dt) that tracks accumulated net change from a fixed start point to a variable endpoint.

Definite integral as accumulated change

The interpretation of ∫_a^b f(t)dt as the net (signed) accumulation of a rate/density f over the interval from a to b.

Integrand (rate/density)

The function inside an integral (e.g., f(t)) representing what is being accumulated, often a rate of change or a density.

Dummy variable

The variable of integration (like t) that is a placeholder; after applying FTC1, it is replaced by the outside variable (like x).

Fundamental Theorem of Calculus, Part 1 (FTC1)

If g(x)=∫_a^x f(t)dt and f is continuous, then g′(x)=f(x).

FTC1 with chain rule (upper limit h(x))

If g(x)=∫_a^{h(x)} f(t)dt, then g′(x)=f(h(x))·h′(x).

Variable lower limit introduces a negative

If g(x)=∫_{h(x)}^a f(t)dt, then g′(x)=−f(h(x))·h′(x) because reversing limits changes the sign.

Both limits depend on x (Leibniz-style rule)

If g(x)=∫_{u(x)}^{v(x)} f(t)dt, then g′(x)=f(v(x))·v′(x) − f(u(x))·u′(x).

Orientation of an integral

The direction from lower to upper limit; switching the order reverses orientation and changes the integral’s sign.

Signed (net) area

The value of ∫_a^x f(t)dt as area above the x-axis minus area below the x-axis over [a,x].

Monotonicity of an accumulation function

For g(x)=∫_a^x f(t)dt, g is increasing where f(x)>0 and decreasing where f(x)<0 because g′(x)=f(x).

Local extrema of an accumulation function

For g(x)=∫_a^x f(t)dt, local maxima/minima occur where f(x)=0 and f changes sign (since g′=f).

Concavity relationship for accumulation functions

If g(x)=∫_a^x f(t)dt, then g″(x)=f′(x), so g is concave up where f is increasing and concave down where f is decreasing.

Misconception: f(x)>0 implies g(x)>0

False in general; g(x)=∫_a^x f(t)dt depends on total accumulated signed area from a to x, including earlier negative/positive contributions.

Linearity of definite integrals

∫a^b (f(x)+g(x))dx = ∫a^b f(x)dx + ∫a^b g(x)dx and ∫a^b c·f(x)dx = c∫_a^b f(x)dx.

Reversing limits property

∫a^b f(x)dx = −∫b^a f(x)dx; also ∫_a^a f(x)dx = 0.

Additivity across intervals

If c is between a and b, then ∫a^b f(x)dx = ∫a^c f(x)dx + ∫_c^b f(x)dx.

Even function (symmetry about y-axis)

A function with f(−x)=f(x); on symmetric limits, ∫{−a}^a f(x)dx = 2∫0^a f(x)dx.

Odd function (origin symmetry)

A function with f(−x)=−f(x); on symmetric limits, ∫_{−a}^a f(x)dx = 0 due to cancellation.

Integral comparison (inequality)

If f(x)≥g(x) for all x in [a,b], then ∫a^b f(x)dx ≥ ∫a^b g(x)dx (in particular, if f≥0 then the integral is ≥0).

Fundamental Theorem of Calculus, Part 2 (FTC2)

If F′(x)=f(x) on [a,b], then ∫a^b f(x)dx = F(b)−F(a) (also written [F(x)]a^b).

Antiderivative

A function F whose derivative is f (F′=f); FTC2 uses an antiderivative to evaluate a definite integral by endpoint subtraction.

Net Change Theorem

If Q′(t)=R(t), then ∫_a^b R(t)dt = Q(b)−Q(a), meaning the integral of a rate equals the net change in the quantity.

Displacement vs. total distance

If velocity v(t) is integrated, ∫ v(t)dt gives displacement (net change in position); total distance requires accounting for sign changes (e.g., integrating |v(t)| or splitting intervals).

Using an accumulation function as an antiderivative

If G(x)=∫a^x f(t)dt, then G′(x)=f(x) so ∫p^q f(x)dx can be found as G(q)−G(p) even without an explicit formula for f.