Revision sheet

Motion with integrals

• Position: start + $\int v$.
• Displacement $\int v$; distance $\int |v|$.

Watch out: split where $v$ changes sign.

Area between curves

1. Find where they meet.
2. Decide which is on top.
3. $\int (\text{top} - \text{bottom})\,dx$.

Watch out: a negative result means the order is reversed.

Area in $y$

• Right minus left, $dy$.
• Limits: the $y$-values where the curves meet.

Watch out: $x$-values are not $y$-limits.

Known cross-sections

1. Side $s(x)$ from the base.
2. Area $A(x)$ of the shape.
3. $V = \int A(x)\,dx$.

Watch out: a diameter is twice the radius.

Disc method

$V = \pi\displaystyle\int_a^b (\text{radius})^2\,dx$.

Watch out: square the radius; keep the $\pi$.

Washer method

$V = \pi\displaystyle\int_a^b (R^2 - r^2)\,dx$, radii from the axis.

Watch out: $R^2 - r^2$, not $(R - r)^2$.