Unit 6: Integration and Accumulation of Change

Integration reverses differentiation and measures accumulation. It ties together antiderivatives, area, net change, and the Fundamental Theorem of Calculus.

Antiderivatives

An antiderivative of \(f\) is any function \(F\) such that

\[F'(x) = f(x).\]

Examples:

\[\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \qquad n \ne -1\] \[\int \frac{1}{x}\,dx = \ln(\lvert x \rvert) + C\] \[\int e^x\,dx = e^x + C\] \[\int \cos x\,dx = \sin x + C\]

Riemann sums

To approximate the accumulation of \(f(x)\) on \([a,b]\), divide the interval into subintervals and sum:

\[\sum_{i=1}^n f(x_i^*)\Delta x\]

where

\[\Delta x = \frac{b-a}{n}.\]

Important choices:

left Riemann sum,
right Riemann sum,
midpoint sum,
trapezoidal approximation.

[Image Placeholder: left, right, midpoint, and trapezoidal approximations on one graph]

Definite integral

The definite integral is the limit of Riemann sums:

\[\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*)\Delta x.\]

Interpretations:

signed area under a curve,
net accumulation,
total change when integrating a rate.

Fundamental Theorem of Calculus

If \(F'(x)=f(x)\), then

\[\int_a^b f(x)\,dx = F(b)-F(a).\]

Also, if

\[g(x) = \int_a^x f(t)\,dt,\]

then

\[g'(x) = f(x)\]

when \(f\) is continuous.

Integrals with variable limits

\[G(x) = \int_{u(x)}^{v(x)} f(t)\,dt,\]

then

\[G'(x) = f(v(x))v'(x) - f(u(x))u'(x).\]

u-substitution

If part of the integrand is the derivative of another part, let

\[u = g(x), \qquad du = g'(x)\,dx.\]

Then

\[\int f(g(x))g'(x)\,dx = \int f(u)\,du.\]

Average value of a function

On \([a,b]\):

\[f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx.\]

Accumulation functions

\[A(x) = \int_a^x f(t)\,dt,\]

then:

\(A'(x)=f(x)\),
\(A\) increases where \(f>0\),
\(A\) decreases where \(f<0\),
critical points of \(A\) occur where \(f=0\) or undefined.

Numerical integration

If exact antiderivatives are unavailable, use:

midpoint rule,
trapezoidal rule,
left/right sums.

Trapezoidal rule with equal spacing \(\Delta x\):

\[\int_a^b f(x)\,dx \approx \frac{\Delta x}{2} \left[y_0 + 2y_1 + 2y_2 + \cdots + 2y_{n-1} + y_n\right].\]

Common mistakes

Forgetting \(+C\) on indefinite integrals.
Using area language when the integral is negative and really means net signed accumulation.
Dropping the chain-rule factor in reverse when using substitution.
Confusing \(\int_a^b f(x)\,dx\) with ordinary multiplication.