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)=$\int_a^x f(t)dt$) that tracks accumulated net change from a fixed start point to a variable endpoint.

Definite integral as accumulated change

The interpretation of $\int_a^b f(t)dt$ 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)=$\int_a^x f(t)dt$ and f is continuous, then g′(x)=$f(x)$.

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

If g(x)=$\int_a^{h(x)} f(t)dt$, then g′(x)=$f(h(x))·h′(x)$.

Variable lower limit introduces a negative

If g(x)=$\int_{h(x)}^a f(t)dt$, 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)=$\int_{u(x)}^{v(x)} f(t)dt$, 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 $\int_a^x f(t)dt$ as area above the x-axis minus area below the x-axis over [a,x].

Monotonicity of an accumulation function

For g(x)=$\int_a^x f(t)dt$, g is increasing where f(x)>0 and decreasing where f(x)

Local extrema of an accumulation function

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

Concavity relationship for accumulation functions

If g(x)=$\int_a^x f(t)dt$, 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)=$\int_a^x f(t)dt$ depends on total accumulated signed area from a to x, including earlier negative/positive contributions.

Linearity of definite integrals

$\int_a^b (f(x)+g(x))dx = \int_a^b f(x)dx + \int_a^b g(x)dx$ and $\int_a^b c·f(x)dx = c\int_a^b f(x)dx$.

Reversing limits property

$\int_a^b f(x)dx = -\int_b^a f(x)dx$; also $\int_a^a f(x)dx = 0$.

Additivity across intervals

If c is between a and b, then $\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$.

Even function (symmetry about y-axis)

A function with f(−x)=$f(x)$; on symmetric limits, $\int_{-a}^a f(x)dx = 2\int_0^a f(x)dx$.

Odd function (origin symmetry)

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

Integral comparison (inequality)

If f(x)≥g(x) for all x in [a,b], then $\int_a^b f(x) dx \geq \int_a^b g(x) dx$ (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 $\int_a^b f(x)dx = F(b)-F(a)$ (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 $\int_a^b R(t)dt = Q(b)−Q(a)$, meaning the integral of a rate equals the net change in the quantity.

Displacement vs. total distance

If velocity v(t) is integrated, $\int v(t)dt$ 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)=$\int_a^x f(t)dt$, then G′(x)=$f(x)$ so $\int_p^q f(x)dx$ can be found as G(q)−G(p) even without an explicit formula for f.