2 Applications of Integration

The modeling idea behind applications

A definite integral is an accumulation tool. If a quantity is built from continuously varying pieces, identify the contribution from one small piece and add all such contributions over the required interval:

The main modeling decisions are the variable of integration, the differential contribution, and the bounds. The differential must have units that combine with the remaining factors to produce the desired final units.

A reliable general process is:

1. Sketch the geometric or physical situation.

2. Choose vertical or horizontal slices according to which description is simpler.

3. Write one representative small contribution, such as $dA$, $dV$, $dW$, or $dF$.

4. Express every factor using the variable of integration.

5. Determine bounds that cover the entire region or physical interval exactly once.

6. Integrate and check the units, sign, magnitude, and limiting behavior.

Takeaway: Most applications become manageable when the small contribution is defined correctly before any integration is performed.

Area between curves

For functions of $x$, vertical slices have width $dx$ and height equal to the upper function minus the lower function. Thus, when $f(x)\ge g(x)$ on $[a,b]$,

For horizontal slices, use right minus left:

The choice is determined by which orientation gives simpler boundaries. If curves cross, split the integral at each intersection so that the subtraction remains nonnegative on each piece.

For example, the curves $y=x$ and $y=x^2$ intersect where

so $x=0$ and $x=1$. On $[0,1]$, the line is above the parabola:

Takeaway: Sketch first, locate intersections, and use top minus bottom or right minus left only on intervals where the order is known.

Volumes of solids

Volume can be found by adding cross-sectional areas. If a slice perpendicular to the $x$-axis has area $A(x)$, then

For rotation about an axis, identify the distances from the axis before selecting a special method. A disk has no hole:

A washer has an outer radius and an inner radius:

Cylindrical shells use slices parallel to the axis. For rotation about the $y$-axis with radius $x$ and height $f(x)$,

Use this selection process:

1. Sketch the region and the axis of rotation.

2. Consider slices perpendicular to the axis; determine whether they form disks or washers.

3. Consider slices parallel to the axis; use shells if their radius and height are simpler.

4. Check that the radii, heights, and bounds describe the intended solid.

For horizontal axes, radii are vertical distances. For vertical axes, radii are horizontal distances, and integrating with respect to $y$ may be more convenient.

Takeaway: The best method is the one that expresses radius, height, or cross-sectional area most directly in the chosen variable.

Arc length and surface area

A curve can be approximated by many short line segments. Applying the distance formula to each segment and taking the limiting sum gives arc length. For $y=f(x)$,

For a curve written as $x=g(y)$,

The underlying differential distance is

When the curve is revolved, each small arc segment sweeps out a thin band. Its area is approximately circumference times arc length:

For $y=f(x)\ge 0$ revolved about the $x$-axis,

For rotation about the $y$-axis, when the radius is the distance $x$,

The radius in a surface-area formula must be a nonnegative distance from the axis. Unlike many area and volume integrals, these integrals frequently require numerical approximation because their antiderivatives are not elementary.

Takeaway: Arc length contributes the factor $ds$; surface area contributes circumference times $ds$.

Average value and work

The average value of a continuous function on $[a,b]$ is its total accumulation divided by the interval length:

For $f(x)=x^2$ on $[0,3]$,

The value is an average over the entire interval, not generally the arithmetic mean of the endpoint values. If $f$ is continuous, the Mean Value Theorem for Integrals guarantees at least one point $c\in[a,b]$ where $f(c)=f_{\text{avg}}$.

For variable-force work, divide the motion into small displacements. If the force at position $x$ is $F(x)$, then

The units are force times distance: joules in the SI system and foot-pounds in the U.S. customary system. If the force opposes the displacement, the work is negative.

For a spring obeying Hooke’s law,

where $k$ is the spring constant. Moving the spring from $x=a$ to $x=b$ requires

Takeaway: Average value divides accumulation by interval length, while work accumulates force over distance.

Lifting, pumping, and fluid force

In lifting and pumping problems, begin with a thin layer. If the layer has volume $dV$, fluid weight density $\delta$, and lifting distance $D(y)$, then

If the layer has cross-sectional area $A(y)$ and thickness $dy$, then $dV=A(y)\,dy$, giving

The lifting distance is measured from the layer’s location to its destination, not simply from the origin.

Fluid pressure increases with depth. For weight density $\delta$, pressure at depth $s$ is

A horizontal strip of a vertical plate at depth $s(y)$, width $w(y)$, and thickness $dy$ has area

Its force is

so the total force is

Use a clearly defined coordinate system and convert the coordinate into depth below the fluid surface before forming the pressure factor.

Takeaway: Pumping integrates weight times lifting distance; hydrostatic force integrates pressure times area.

Mass, moments, and center of mass

Integration also models distributed mass and its location. For a thin rod on $[a,b]$ with linear density $\rho(x)$,

For a lamina under $y=f(x)$, above the $x$-axis, between $x=a$ and $x=b$, with constant density $\rho$, the mass is density times area:

The moments about the coordinate axes are

and

The center of mass is then

The factor $x$ in $M_y$ weights mass according to horizontal distance from the $y$-axis. Similarly, the squared-function expression in $M_x$ results from weighting horizontal strips by their vertical distances from the $x$-axis.

Takeaway: Mass integrates density, moments integrate distance-weighted density, and the center of mass is obtained by dividing each moment by total mass.

Checking an integral model

Before accepting an answer, perform checks that reflect the original model.

• Geometry: The integrand should represent a nonnegative length, area, volume, or physical contribution when the situation requires one.

• Bounds: The bounds should cover the entire region or motion exactly once.

• Distances: Radii, depths, and lifting distances must be measured from the correct reference axis, surface, or destination.

• Units: Area should have square units, volume cubic units, arc length units of length, work units of force times distance, and force units of force.

• Signs: A negative answer may indicate opposing work or signed accumulation, but it can also signal a reversed subtraction order or an incorrect distance.

• Limiting behavior: If a height, density, force, or interval shrinks to zero, the modeled total should respond accordingly.

A useful final question is: does the differential expression describe one small piece of exactly the quantity requested? If not, revise the model before evaluating the integral.

Takeaway: In applications of integration, a sound setup and dimensional check are often more important than the final antiderivative.

• Slicing method

• Washer method

• Surface area of revolution