Unit 6 Integration Notes: Connecting Accumulation, Area, and Antiderivatives

Fundamental Theorem of Calculus (FTC)

The pair of theorems that connect differentiation and definite integration: Part 1 gives the derivative of accumulation functions, and Part 2 evaluates definite integrals using antiderivatives.

Accumulation function

A function defined by an integral with a variable limit, e.g., $F(x)=\int_a^x f(t)dt$, representing net accumulated change from a to $x$.

Rate function (integrand)

The function being accumulated in an integral (often interpreted as a rate), e.g., $f(t)$ in $\int_a^x f(t)dt$.

Variable of integration (dummy variable)

The placeholder variable inside the integral (e.g., $t$ in $\int f(t)dt$) whose name does not affect the integral’s value.

Net accumulated change

The total signed accumulation from a to $b$ given by $\int_a^b f(x)dx$; includes both increases (positive) and decreases (negative).

Signed area (net area)

Area above the x-axis counted positive and below the x-axis counted negative in a definite integral.

Displacement

Change in position; if $v(t)$ is velocity, then $\int_a^b v(t)dt$ gives displacement over $[a,b]$.

FTC Part 1

If f is continuous and F(x)=$\int_a^x f(t)dt$, then F'(x)=f(x).

FTC Part 1 with chain rule (upper limit g(x))

If $G(x)=\int_a^{g(x)} f(t)dt$, then $G'(x)=f(g(x))·g'(x)$.

Leibniz rule (both limits vary)

If $H(x)=\int_{h(x)}^{g(x)} f(t)dt$, then $H'(x)=f(g(x))g'(x) − f(h(x))h'(x)$.

Critical point of an accumulation function

A point $c$ where $A'(c)=0$; for $A(x)=\int_a^x f(t)dt$, this occurs where $f(c)=0$ (an extremum is not guaranteed).

Local maximum of an accumulation function

Occurs where $A'(x)$ changes from positive to negative; for $A'(x)=f(x)$, this is where $f$ changes from $+$ to $−$.

Local minimum of an accumulation function

Occurs where $A'(x)$ changes from negative to positive; for $A'(x)=f(x)$, this is where $f$ changes from $−$ to $+$.

Concavity of an accumulation function

For $A(x)=\int_a^x f(t)dt$ with differentiable $f$, $A''(x)=f'(x)$; $A$ is concave up where $f$ is increasing and concave down where $f$ is decreasing.

Total distance traveled

For velocity $v(t)$, total distance is $\int_a^b |v(t)|dt$ (not $\int_a^b v(t)dt$, which gives displacement).

Definite integral

A number $\int_a^b f(x)dx$ representing net accumulation (signed area) from a to b.

Linearity of integrals

Properties: $\int_a^b (f+g)dx=\int_a^b fdx+\int_a^b gdx$ and $\int_a^b c·f dx = c\int_a^b f dx$.

Reversing bounds property

Switching limits changes the sign: $\int_a^b f(x)dx = −\int_b^a f(x)dx$.

Zero-width interval property

An integral over identical bounds is zero: $\int_a^a f(x)dx = 0$.

Additivity over intervals

Breaking at an interior point $c$: $\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$.

Comparison property (integrals)

If $f(x)≥g(x)$ on $[a,b]$, then $\int_a^b f(x)dx ≥ \int_a^b g(x)dx$ (in particular, if $f≥0$ then the integral is ≥0).

Even function symmetry (integrals)

If f is even (f(−x)=f(x)), then $\int_{-a}^a f(x)dx = 2\int_0^a f(x)dx$.

Odd function symmetry (integrals)

If $f$ is odd ($f(−x)=−f(x)$), then $\int_{−a}^a f(x)dx = 0$.

FTC Part 2

If $f$ is continuous on $[a,b]$ and $F' = f$, then $\int_a^b f(x)dx = F(b) − F(a)$.

Evaluation notation (bracket notation)

The shorthand $[F(x)]_a^b$ meaning $F(b) − F(a)$ when evaluating a definite integral via an antiderivative.