← Unit 8

Unit 8 Cheat Sheet

Every formula in Applications of Integration — and, more importantly, when to use which. The methods that look alike (disc vs. washer, dx vs. dy, displacement vs. distance) are exactly where points are lost, so each block ends with the decision that picks the right tool.

Average Value 8.1

The mean height of a function over an interval.

average value

MVT for Integrals: if \(f\) is continuous, some \(c\) in \([a,b]\) actually hits this average value.

trap

Divide by \(b-a\). The integral alone is not the average.

Motion 8.2

Velocity integrates to position; signs carry meaning.

displacement (signed)

total distance

position from velocity

Displacement or distance?

net change→ displacement, \(\int v\) (keep signs)

total traveled→ distance, \(\int|v|\) — find where \(v=0\) and split

Speeding up or slowing down?

v, a same sign→ speeding up

v, a opposite→ slowing down

Accumulation 8.3

Integrate a rate to get a total; the FTC links them.

fundamental theorem (part 2)

Net change: \(\int_a^b(\text{rate})\,dt\) = total change. Amount: start value \(+\) net change.

Chain rule: if the upper limit is \(u(x)\), then \(\frac{d}{dx}\int_a^{u}f\,dt=f(u)\cdot u'\).

trap

Add the initial amount. The integral gives the change, not the final total.
\(F'(x)\) is the integrand evaluated at \(x\) — don't integrate, and don't differentiate it again.

Before you integrate every problem

A 20-second checklist that catches most lost points.

Always

sketch first→ see the region, the axis, and which curve is on top

find bounds→ set curves equal; check for every intersection

pick the variable→ strips ⊥ x-axis use \(dx\); ⊥ y-axis use \(dy\)

Units tell: area is square units, volume cubic, arc length linear. If your setup's units don't match, the setup is wrong.

Calculator section? Set up the exact integral, then evaluate numerically — you still earn the setup points even if the antiderivative is ugly.

Area Between Curves 8.4 – 8.6

Integrate the gap between the boundaries.

in x (vertical strips)

in y (horizontal strips)

Integrate in x or y?

strips ⊥ x-axis→ \(dx\), curves as \(y=f(x)\), top − bottom

strips ⊥ y-axis→ \(dy\), curves as \(x=g(y)\), right − left

sideways region→ \(dy\) avoids splitting (e.g. \(x=y^2\))

trap

Find every intersection. Curves can cross more than once — the top/bottom may swap.

Volume by Cross-Sections 8.7 – 8.8

Stack up slices of a known shape on a base region.

volume (any shape)

Slice area \(A\), where \(s\) = the base region's width:

\(s\) = top − bottom (or right − left). The rectangle here uses height = twice the base (\(h=2s\)), giving \(2s^2\) — but that multiplier changes per problem, so read it. For semicircles, \(s\) is the diameter, so radius \(=s/2\).

trap

No \(\pi\) unless it's a circle. Squares/triangles get their own area formula.
Slices ⊥ y-axis ⇒ integrate in \(y\).

Solids of Revolution 8.9 – 8.12

Spin a region around an axis; slices are circles.

disc (no hole)

washer (with hole)

R = outer radius (axis → far edge); r = inner radius (axis → near edge).

Disc or washer?

region touches axis→ disc (no gap, \(r=0\))

gap to the axis→ washer, \(R^2-r^2\)

axis is a line \(y=k\)→ measure each radius from that line

trap

\(R^2-r^2\), not \((R-r)^2\) — subtract the squares.
Square the whole radius: \((x^2)^2=x^4\).

Arc Length 8.13 · BC only

A curve's length = the sum of tiny hypotenuses. Pythagoras, integrated.

y = f(x)

x = g(y)

The perfect-square trick. The integrand is almost never integrable by hand — exam problems are rigged so the inside becomes a perfect square:

trap

Square the derivative before adding 1: \((2x)^2=4x^2\).
If it's not a perfect square, just set up the integral — that earns the credit.

∫

Half of Unit 8 is one idea: an integral adds up infinitely many tiny pieces into a total — a rate into a net change, strips into an area, discs into a volume, hypotenuses into a length. Set up the right piece, and the integral does the rest.

Test yourself on the exam set →

Accumulation Area Cross-Sections Solids Arc Length