Revision sheet

Distance from velocity

• Area under $v$ = distance (when $v \ge 0$).
• Signed area (below the axis negative) = displacement.

Watch out: use the right area formula, ½ for triangles.

Riemann sums

• $\Delta x = (b - a)/n$; add height × $\Delta x$.
• Left, right or midpoint heights.
• Increasing $f$: left under, right over.

Watch out: multiply by the width.

The definite integral

• Signed area: below the axis is negative.
• Reverse the limits: change the sign. Adjacent intervals add.
• $\displaystyle\int_a^b c\,dx = c(b - a)$.

Watch out: a constant is integrated over the whole length.

Antiderivatives

• $x^n \to \dfrac{x^{n+1}}{n+1}$; $\tfrac{1}{x} \to \ln|x|$; $e^x \to e^x$; $\cos \to \sin$; $\sin \to -\cos$.
• Use a given value to find $C$.

Watch out: check by differentiating.

Fundamental Theorem (evaluation)

$\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)$, with $F' = f$.

Watch out: upper minus lower; check the antiderivative's signs.

Average value

$f_{\text{avg}} = \dfrac{1}{b - a}\displaystyle\int_a^b f(x)\,dx$.

Watch out: divide by the length; do not just average the end values.

Accumulation functions

• $\dfrac{d}{dx}\displaystyle\int_a^x f(t)\,dt = f(x)$.
• Upper limit $u(x)$: multiply by $u'(x)$.
• $g$ rises where $f > 0$, falls where $f < 0$.

Watch out: the answer is the integrand, not its antiderivative.

Net change

$F(b) = F(a) + \displaystyle\int_a^b F'(t)\,dt$: start + accumulated change.

Watch out: a rate at one moment is not an amount.