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)=∫_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 ∫_a^x f(t)dt.

Variable of integration (dummy variable)

The placeholder variable inside the integral (e.g., t in ∫ 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 ∫_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 ∫_a^b v(t)dt gives displacement over [a,b].

FTC Part 1

If f is continuous and F(x)=∫_a^x f(t)dt, then F'(x)=f(x).

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

If G(x)=∫_a^{g(x)} f(t)dt, then G'(x)=f(g(x))·g'(x).

Leibniz rule (both limits vary)

If H(x)=∫_{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)=∫_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)=∫_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 ∫a^b |v(t)|dt (not ∫a^b v(t)dt, which gives displacement).

Definite integral

A number ∫_a^b f(x)dx representing net accumulation (signed area) from a to b.

Linearity of integrals

Properties: ∫a^b (f+g)dx=∫a^b fdx+∫a^b gdx and ∫a^b c·f dx = c∫_a^b f dx.

Reversing bounds property

Switching limits changes the sign: ∫a^b f(x)dx = −∫b^a f(x)dx.

Zero-width interval property

An integral over identical bounds is zero: ∫_a^a f(x)dx = 0.

Additivity over intervals

Breaking at an interior point c: ∫a^b f(x)dx = ∫a^c f(x)dx + ∫_c^b f(x)dx.

Comparison property (integrals)

If f(x)≥g(x) on [a,b], then ∫a^b f(x)dx ≥ ∫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 ∫{−a}^a f(x)dx = 2∫0^a f(x)dx.

Odd function symmetry (integrals)

If f is odd (f(−x)=−f(x)), then ∫_{−a}^a f(x)dx = 0.

FTC Part 2

If f is continuous on [a,b] and F' = f, then ∫_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.