Applications of Integration

The average value 平均值 of a continuous function $f$ over $[a,b]$ is the integral divided by the interval length:

It is the constant height of a rectangle on $[a,b]$ with the same area as under $f$. Do not confuse it with the average rate of change (which divides change in $f$ by the interval), or with a Riemann-sum average of a few values – the exam distinguishes these carefully. Report units in context (e.g. the average rate "in vehicles per hour").

Worked example. The average value of $f(x)=x^2$ on $[0,3]$ is

Integration reverses the motion links of Unit 4. For a particle in straight-line motion over $[t_1,t_2]$:

• Displacement 位移 (net change in position) $= \displaystyle\int_{t_1}^{t_2} v(t)\,dt$.
• Total distance travelled 总路程 $= \displaystyle\int_{t_1}^{t_2} |v(t)|\,dt$ – integrate speed, so direction changes add up instead of cancelling.
• Position at a later time $= x(t_1) + \displaystyle\int_{t_1}^{t} v(s)\,ds$; velocity from acceleration $= v(t_1)+\int a$.

The displacement-versus-distance distinction (with the absolute value) is a classic graded point.

A function defined as an integral accumulates a rate of change. The net change 净变化 theorem is the everyday tool: the definite integral of a rate over an interval gives the net change of the quantity:

So "how much water is in the tank at $t=5$" = starting amount + $\int_0^5(\text{inflow}-\text{outflow})\,dt$. Watch signs (in minus out) and units. Many multi-part FRQs are built entirely on this idea – often paired with "write, but do not evaluate, an integral expression that gives the total...".

The area between two curves $y=f(x)$ (top) and $y=g(x)$ (bottom) on $[a,b]$ is:

Find the intersection points first to set the limits, and always subtract bottom from top.

Worked example. Find the area between $y=x$ and $y=x^2$. They cross at $x=0$ and $x=1$, and on $(0,1)$ the line $y=x$ is on top, so

The shaded region runs from one intersection $a$ to the next $b$; its area is $\int_a^b(\text{top}-\text{bottom})\,dx$.

Some regions are easier described with horizontal slices – integrate with respect to $y$:

Choose $x$- or $y$-slices to keep a single top/bottom (or right/left) pair over the whole region.

If the curves cross more than twice, one function is on top for part of the region and the other on top elsewhere. Split into a sum of integrals at each crossing, or integrate the absolute value of the difference:

Determine which curve is higher on each subinterval before writing the integrals.

If a solid has a known cross section 横截面 perpendicular to an axis, its volume is the integral of the cross-sectional area:

For square cross sections with side equal to the region's height $s(x)=f(x)-g(x)$, use $A(x)=[s(x)]^2$; for a rectangle of height $k\cdot s(x)$, use $A(x)=k\,[s(x)]^2$. The exam phrase "the base of a solid... cross sections are squares/rectangles" signals exactly this.

Same method, different area formula: for an equilateral triangle 三角形 of side $s$, $A=\tfrac{\sqrt{3}}{4}s^2$; for a semicircle 半圆 of diameter $s$, $A=\tfrac{\pi}{8}s^2$. Set the shape's key length equal to the region's slice length $s(x)$, then integrate $A(x)$.

Revolving a region around an axis makes a solid of revolution 旋转体. If the region touches the axis, each slice is a disc 圆盘 of radius $r$ = the function value:

Around the $x$-axis, $r=f(x)$ and integrate in $x$; around the $y$-axis, express $x$ as a function of $y$ and integrate in $y$.

The disc method: rotating y=f(x) about the axis sweeps out disks of radius f(x)

Worked example. Revolve the region under $y=\sqrt{x}$ from $x=0$ to $x=4$ about the $x$-axis. Each disc has radius $r=\sqrt{x}$, so

Rotating a region about an axis sweeps out a solid of revolution

Around any horizontal or vertical line $y=k$ or $x=k$, the radius is the distance from the curve to that line, e.g. $r(x)=|f(x)-k|$. Set up the radius carefully, then use $V=\pi\int r^2$.

When the region does not touch the axis, each slice is a washer 垫圈 (a ring) with an outer radius $R$ and inner radius $r$:

$R$ is the distance from the axis to the farther boundary and $r$ to the nearer one. Around a line other than an axis, both radii are measured as distances to that line – a very common exam variation (e.g. revolving about $y=-2$).

• Area between curves is $\int(\text{top}-\text{bottom})\,dx$ — find the intersection points for the limits and keep top minus bottom.
• For a volume of revolution, add up disc/washer cross-sections of area $\pi r^2$ (or $\pi(R^2-r^2)$).
• The average value of $f$ on $[a,b]$ is $\tfrac{1}{b-a}\int_a^b f\,dx$.
• Accumulated change is $\int$ of a rate: total = initial value $+\int_a^b(\text{rate})\,dt$.
• Integration means "adding up infinitely many tiny pieces" — set up the integrand as one thin slice.