Applications of Integration - Theory & Concepts

Applications of Integration

$Learning Objectives$

Compute geometric area when bounding curves exchange order.
Calculate volumes by slicing, disks, washers, shells, and Pappus' centroid theorem.
Determine arc length and surface area using distance-based radii.
Compute centroids, first moments, and second moments of area for planar regions.
Distinguish geometric second moment of area from mass moment of inertia.
Model variable-force work, pumping work, hydrostatic force, and center of pressure.
Apply definite integrals to economic surplus and continuous probability.

Applications of integration share one principle: model a quantity on a small element, then sum those elements by integration. Correct setup depends on geometry, units, sign, and the physical meaning of the differential element.

Area Between Curves

$Geometric Area$

For continuous curves $y=f(x)$ and $y=g(x)$ ,

A=\int_a^b |f(x)-g(x)|\,dx.

If one curve is known to stay above the other on the whole interval, the absolute value may be replaced by “top minus bottom.” If the curves cross, split the integral at their intersection points or retain the absolute value.

Area Between Curves

a = -1.0b = 2.0

f(x)=x; g(x)=x²

Numerical geometric area: 1.83333

The simulation integrates |f−g|. When the curves exchange order, the shaded band remains physical area rather than being clipped to zero.

$Integrating with Respect to y$

When horizontal slices are simpler, use “right minus left”:

A=\int_c^d |x_{\rm right}(y)-x_{\rm left}(y)|\,dy.

The bounds must be $y$ -values.

Polar Area and Arc Length

Polar Area

Area swept by a polar radius.

A=\frac12\int_\alpha^\beta r(\theta)^2\,d\theta.

Polar Arc Length

Length of a polar curve.

L=\int_\alpha^\beta\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}\,d\theta.

Volumes by Slicing

$General Cross-Section Formula$

If the area of the cross section perpendicular to the integration axis is $A(x)$ ,

V=\int_a^b A(x)\,dx.

Volume by Slicing: Midpoint Approximation

Number of slices: 8

V=\pi\int_0^1x^2dx=\pi/3

Midpoint sum

1.0431069

Exact volume

1.0471976

Relative error

0.3906%

The bars are a qualitative slice visualization; the reported midpoint sum and error are quantitative. Increasing n should drive the numerical approximation toward the exact integral.

Disk and Washer Methods

Disk Method

Solid cross sections perpendicular to the axis.

V=\pi\int_a^b R(x)^2\,dx.

Washer Method

Annular cross sections perpendicular to the axis.

V=\pi\int_a^b\left[R(x)^2-r(x)^2\right]dx,

where $R(x)\ge r(x)\ge0$ are distances from the axis of revolution.

Solid of Revolution: Model and Formula Must Match

y=\sqrt{x},\ 0\le x\le3,\qquad V=\pi\int_0^3(\sqrt{x})^2dx=\frac{9\pi}{2}

The visual uses the exact functions and interval shown in the displayed integral. No hidden rescaling is applied to the mathematical model.

Cylindrical Shells

Shell Method

Thin shells parallel to the axis of revolution.

V=2\pi\int_a^b \rho(x)h(x)\,dx,

where $\rho(x)$ is the nonnegative distance from the slice to the axis and $h(x)$ is the shell height.

$Radius Means Distance$

For rotation about a vertical line $x=c$ , the shell radius is $\rho(x)=|x-c|$ , not the signed coordinate $x-c$ . For rotation about a horizontal line $y=c$ , use the corresponding vertical distance.

Arc Length

Cartesian Arc Length

Length of y=f(x).

L=\int_a^b\sqrt{1+[f'(x)]^2}\,dx.

Arc Length: Polygonal Approximation

L=\int_0^1\sqrt{1+4x^2}\,dx=\frac{\sqrt5}{2}+\frac14\operatorname{asinh}(2)

Number of chords: 8

Polygonal approximation

1.47777799

Analytical arc length

1.47894286

Absolute error

1.165e-3

This finite chord sum is an approximation in model coordinates, not an exact length and not a Riemann sum. Its limit motivates the arc-length integral.

Surface Area of Revolution

$General Surface Formula$

If a smooth curve is revolved about an axis that does not pass through the generating arc, the lateral surface area is

S=2\pi\int r\,ds,

where $r$ is the nonnegative perpendicular distance from the curve element to the axis.

About a Horizontal Axis y=c

Curve y=f(x).

S=2\pi\int_a^b |f(x)-c|\sqrt{1+[f'(x)]^2}\,dx.

About a Vertical Axis x=c

Curve y=f(x).

S=2\pi\int_a^b |x-c|\sqrt{1+[f'(x)]^2}\,dx.

$Do Not Use a Signed Radius$

A radius is a distance. If $f(x)-c$ or $x-c$ changes sign, split the interval or retain the absolute value. A signed radius can produce cancellation and a nonphysical negative surface contribution.

Centroids and First Moments of Area

For a uniform planar region, the centroid is a geometric property. Density is not required. If a nonuniform density is introduced, the corresponding center of mass becomes a mass property.

$Region Between Two Curves$

For $f(x)\ge g(x)$ on $[a,b]$ ,

A=\int_a^b[f-g]dx,

M_y=\int_a^b x[f-g]dx, \qquad M_x=\frac12\int_a^b[f^2-g^2]dx,

\bar x=\frac{M_y}{A},\qquad \bar y=\frac{M_x}{A}.

Second Moment of Area

$Second Moment of Area$

For a planar area $A$ ,

I_x=\int_A y^2\,dA,\qquad I_y=\int_A x^2\,dA.

These are geometric section properties with units of length to the fourth power, such as $\mathrm{mm}^4$ or $\mathrm{m}^4$ . In structural mechanics they appear in bending stiffness $EI$ , bending stress relations, and deflection calculations.

$Between y=f(x) and y=g(x)$

For $f(x)\ge g(x)$ ,

I_x=\frac13\int_a^b\left[f(x)^3-g(x)^3\right]dx,

I_y=\int_a^b x^2[f(x)-g(x)]dx.

$Do Not Confuse Area and Mass Moments$

The mass moment of inertia is

I_m=\int r^2\,dm,

with units of mass times length squared, and it enters rotational dynamics through $\sum M=I_m\alpha$ . The second moment of area uses $dA$ , has units $L^4$ , and does not by itself describe resistance to angular acceleration.

Work by a Variable Force

Variable-Force Work

One-dimensional work along x.

W=\int_a^b F(x)\,dx.

$Hooke's Law$

For an ideal spring, $F=kx$ , so stretching from $x=a$ to $x=b$ requires

W=\int_a^b kx\,dx=\frac{k}{2}(b^2-a^2).

Using one constant endpoint force generally gives the wrong work because the spring force varies with extension.

Pumping Fluids

For a thin fluid slice, model

dW=(\text{weight density})(\text{slice volume})(\text{lifting distance}).

If mass density is $\rho$ , weight density is $\rho g$ .

Hydrostatic Force and Center of Pressure

Hydrostatic Pressure

Gauge pressure at depth h below a free surface.

p=\rho gh.

Hydrostatic Force

Force on a plane surface by integration.

F=\int_A p\,dA=\rho g\int_A h\,dA.

$Center of Pressure$

For a vertical plane surface, the depth of the resultant force satisfies

h_{cp}=\frac{\int_A h^2\,dA}{\int_A h\,dA}.

For a vertical rectangle whose top edge is at the free surface and height is $H$ , $h_{cp}=2H/3$ , deeper than the centroid depth $H/2$ because pressure increases with depth.

Theorems of Pappus–Guldinus

The centroid theorems apply when the generating curve or planar region is revolved through a full turn about an external coplanar axis that does not intersect the generating curve/region (apart from standard limiting boundary cases handled separately).

$Pappus Formulas$

Surface generated by a plane curve of length $L$ :

S=(2\pi\bar r)L.

Volume generated by a plane area $A$ :

V=(2\pi\bar r)A.

Here $\bar r$ is the perpendicular distance from the generating curve/area centroid to the axis.

Pappus Volume Theorem: External-Axis Condition

Generating-circle radius r = 1.5Centroid distance to axis d = 3.0

Validity: Valid external axis: d > r

d=3.0,\quad r=1.5,\quad d>r

V=(2\pi d)(\pi r^2)=2\pi^2dr^2

Volume: 133.2397 cubic units

Both controls enforce the theorem condition. Increasing r automatically moves d outward if necessary, so the displayed Pappus result never describes an axis cutting through the generating disk.

Economic Surplus

$Consumer and Producer Surplus$

If market quantity is $Q^*$ and equilibrium price is $P^*$ ,

CS=\int_0^{Q^*}[D(q)-P^*]dq,

PS=\int_0^{Q^*}[P^*-S(q)]dq,

provided demand lies above the market price and supply lies below it over the traded interval.

Continuous Probability

$Probability Density Function$

A continuous PDF satisfies $f(x)\ge0$ and

\int_{-\infty}^{\infty}f(x)\,dx=1.

Then

P(a\le X\le b)=\int_a^b f(x)\,dx.

A density value itself is not a point probability; for a continuous random variable, $P(X=x)=0$ .

$Dimensional Check$

Area has units $L^2$ , volume $L^3$ , arc length $L$ , surface area $L^2$ , second moment of area $L^4$ , mass moment of inertia $ML^2$ , force $ML/T^2$ , and work $ML^2/T^2$ . Unit consistency is a powerful setup check.

Previous TopicTechniques of Integration - Worked Examples

Quiz Me

Next TopicApplications of Integration - Worked Examples

Prev Next

Quiz Me