Applications of Integrals

The One Thing

Definite integration calculates geometric area by summing infinitely many thin strips. The essential procedure is to identify the region, determine its boundaries and limits, select vertical or horizontal strips, and ensure that signed integrals are converted into non-negative geometric areas.

Definitions and Results

• Definite Integral: The value of an integral between two limits, representing an accumulated quantity such as area. Using an antiderivative $F(x)$,

$${\int }_a^{b}f(x)dx=F(b)-F(a).$$

• Area Under a Curve: The region between $y=f(x)$, the $x$-axis, and the vertical lines $x=a$ and $x=b$.

• Area Above the $x$-axis: If $f(x)\ge 0$ on $[a,b]$,

$$\text{Area}={\int }_a^{b}f(x)dx.$$

• Area Below the $x$-axis: If $f(x)\le 0$ on $[a,b]$, the geometric area is

$$-{\int }_a^{b}f(x)dx,$$ or equivalently $${\int }_a^b|f(x)|dx.$$

• Geometric Area: The actual non-negative size of a region. It differs from a signed integral, which may be negative below the $x$-axis.

• Area Between Two Curves: If $y=f(x)$ lies above $y=g(x)$ on $[a,b]$,

$$\text{Area}={\int }_a^b[f(x)-g(x)]dx.$$

• Points of Intersection: Points where two curves meet, found by solving

$$f(x)=g(x).$$ These points commonly determine the limits of integration.

• Upper and Lower Curves: For vertical strips, the upper curve has the greater $y$-value and the lower curve has the smaller $y$-value on the selected interval.

• Horizontal Strip Method: If the curves are written as $x=f(y)$ and $x=g(y)$, and $x=f(y)$ is the right boundary while $x=g(y)$ is the left boundary, then

$$\text{Area}={\int }_c^d[f(y)-g(y)]dy.$$

• Symmetry: If a region is symmetric about the $y$-axis, its total area may be found by doubling the area on one side of the $y$-axis.

• Area Between a Curve and the $x$-axis: In general,

$$\text{Area}={\int }_a^b|f(x)|dx.$$ If the curve crosses the $x$-axis, the integral must be split at the crossing points.

• Standard Antiderivatives:

$$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C,n\ne -1,$$ $$\int \frac{1}{x}dx=\ln |x|+C,$$ $$\int e^{x}dx=e^x+C.$$

• Units: A geometric area is expressed as a non-negative quantity, commonly in square units.

Worked Methods

1. Area Under a Curve Above the $x$-axis

• Sketch the curve and identify the interval $[a,b]$.
• Confirm that $y=f(x)$ is non-negative throughout the interval.
• Use vertical strips of width $dx$. The strip height is $f(x)$.
• Integrate:

$$\text{Area}={\int }_a^{b}f(x)dx.$$

• Evaluate the definite integral using

$${\int }_a^{b}f(x)dx=F(b)-F(a).$$

2. Area Below the $x$-axis

• Sketch the curve and identify the interval $[a,b]$.
• Confirm that $f(x)\le 0$ throughout the interval.
• Calculate the signed integral:

$${\int }_a^{b}f(x)dx.$$

• Convert it to geometric area:

$$\text{Area}=-{\int }_a^{b}f(x)dx.$$ Equivalently, $$\text{Area}={\int }_a^b|f(x)|dx.$$

3. Area When a Curve Crosses the $x$-axis

• Sketch the region.
• Find the $x$-coordinates where the curve crosses the $x$-axis by solving $f(x)=0$.
• Divide the interval at each crossing point.
• Determine whether the function is above or below the $x$-axis on each subinterval.
• Integrate each part as a positive area:

$$\text{Area}=\int |f(x)|dx,$$ or use separate integrals with a negative sign for portions below the $x$-axis.

• Add the resulting non-negative areas.

The area between $y=f(x)$ and the $x$-axis is therefore not always equal to the value of a single signed integral when the curve crosses the axis.

4. Area Between Two Curves Using Vertical Strips

• Sketch both curves and the enclosed region.
• Find their points of intersection by solving

$$f(x)=g(x).$$

• Use the intersection $x$-coordinates, together with any other boundary lines, as limits.
• Determine which curve is upper and which is lower on the interval.
• A thin vertical strip has approximate area

$$[\text{upper function}-\text{lower function}]dx.$$

• Integrate:

$$\text{Area}={\int }_a^b[f(x)-g(x)]dx$$ when $f(x)$ is above $g(x)$.

• If the upper or lower curve changes, split the integral at the point where the order changes.

5. Finding Intercepts for Regions Bounded by Coordinate Axes

• For an $x$-intercept, put $y=0$ and solve for $x$.
• For a $y$-intercept, put $x=0$ and solve for $y$.
• Use the intercepts, the coordinate axes, and the curve to identify the enclosed region.
• Choose the integration variable and establish the appropriate limits.
• Evaluate the resulting area integral.

6. Area Using Horizontal Strips

• Rewrite the boundaries in the form

$$x=f(y),x=g(y),$$ where necessary.

• Sketch the region and identify its lower and upper $y$-limits, $c$ and $d$.
• Determine the right and left boundaries.
• A thin horizontal strip has approximate area

$$[\text{right function}-\text{left function}]dy.$$

• If $x=f(y)$ is the right boundary and $x=g(y)$ is the left boundary, integrate:

$$\text{Area}={\int }_c^d[f(y)-g(y)]dy.$$

7. Using Symmetry

• Establish from the diagram or equations that the region is symmetric about the $y$-axis.
• Calculate the area on one side of the $y$-axis.
• Double that result:

$$\text{Total area}=2(\text{area on one side}).$$

8. General Procedure for an Area Problem

• Sketch the region.
• Find intersections and intercepts.
• Identify all relevant boundary lines or curves.
• Choose $dx$ or $dy$, depending on whether vertical or horizontal strips give the simpler description.
• Identify the upper and lower boundaries for vertical strips, or the right and left boundaries for horizontal strips.
• Split the integral if curves intersect, the boundary order changes, or the curve crosses the $x$-axis.
• Evaluate the integral.
• Check that the final answer is non-negative and is expressed in square units.

Where It Goes Wrong

• Using a signed integral as the geometric area when the curve lies below the $x$-axis or crosses it; portions below the axis must be made positive.
• Failing to solve $f(x)=g(x)$ when two curves define the region, leading to incorrect limits of integration.
• Reversing the upper and lower curves for vertical strips, or the right and left boundaries for horizontal strips.
• Using one integral when the upper or lower curve changes, rather than splitting at the relevant intersection.
• Choosing $dx$ or $dy$ without considering which direction gives a single, clear boundary description.
• Omitting intercepts found by putting $x=0$ or $y=0$ when coordinate axes form part of the boundary.

What Gets Asked

This material supports questions requiring students to:

• Calculate the area under a curve above or below the $x$-axis.
• Find the geometric area when a curve crosses the $x$-axis.
• Determine the intersections of two curves and use them as integration limits.
• Calculate the area between two curves using upper minus lower.
• Identify and use intercepts when a region is bounded by a curve and the coordinate axes.
• Decide whether vertical strips $dx$ or horizontal strips $dy$ are more appropriate.
• Apply the horizontal strip method using right boundary minus left boundary.
• Use symmetry to reduce the amount of integration required.
• Evaluate definite integrals using standard antiderivatives.
• Sketch a region and set up an integral with correct boundaries, limits, signs, and square-unit interpretation.