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Essential Concepts

Areas Between Curves

• Just as definite integrals can be used to find the area under a curve, they can also be used to find the area between two curves.
• To find the area between two curves defined by functions, integrate the difference of the functions.
• If the graphs of the functions cross, or if the region is complex, use the absolute value of the difference of the functions. In this case, it may be necessary to evaluate two or more integrals and add the results to find the area of the region.
• Sometimes it can be easier to integrate with respect to [latex]y[/latex] to find the area. The principles are the same regardless of which variable is used as the variable of integration.

Determining Volumes by Slicing

• Definite integrals can be used to find the volumes of solids. Using the slicing method, we can find a volume by integrating the cross-sectional area.
• For solids of revolution, the volume slices are often disks and the cross-sections are circles. The method of disks involves applying the method of slicing in the particular case in which the cross-sections are circles, and using the formula for the area of a circle.
• If a solid of revolution has a cavity in the center, the volume slices are washers. With the method of washers, the area of the inner circle is subtracted from the area of the outer circle before integrating.

Volumes of Revolution: Cylindrical Shells

• The method of cylindrical shells is another method for using a definite integral to calculate the volume of a solid of revolution. This method is sometimes preferable to either the method of disks or the method of washers because we integrate with respect to the other variable. In some cases, one integral is substantially more complicated than the other.
• The geometry of the functions and the difficulty of the integration are the main factors in deciding which integration method to use.

Arc Length of a Curve and Surface Area

• The arc length of a curve can be calculated using a definite integral.
• The arc length is first approximated using line segments, which generates a Riemann sum. Taking a limit then gives us the definite integral formula. The same process can be applied to functions of [latex]y.[/latex]
• The concepts used to calculate the arc length can be generalized to find the surface area of a surface of revolution.
• The integrals generated by both the arc length and surface area formulas are often difficult to evaluate. It may be necessary to use a computer or calculator to approximate the values of the integrals.

Key Equations

• Area between two curves, integrating on the [latex]x[/latex]-axis
  [latex]A={\displaystyle\int }_{a}^{b}\left[f(x)-g(x)\right]dx[/latex]
• Area between two curves, integrating on the [latex]y[/latex]-axis
  [latex]A={\displaystyle\int }_{c}^{d}\left[u(y)-v(y)\right]dy[/latex]
• Disk Method along the [latex]x[/latex]-axis
  [latex]V={\displaystyle\int }_{a}^{b}\pi {\left[f(x)\right]}^{2}dx[/latex]
• Disk Method along the [latex]y[/latex]-axis
  [latex]V={\displaystyle\int }_{c}^{d}\pi {\left[g(y)\right]}^{2}dy[/latex]
• Washer Method
  [latex]V={\displaystyle\int }_{a}^{b}\pi \left[{(f(x))}^{2}-{(g(x))}^{2}\right]dx[/latex]
• Method of Cylindrical Shells
  [latex]V={\displaystyle\int }_{a}^{b}(2\pi xf(x))dx[/latex]
• Arc Length of a Function of [latex]x[/latex]
  [latex]\text{Arc Length}={\displaystyle\int }_{a}^{b}\sqrt{1+{\left[{f}^{\prime }(x)\right]}^{2}}dx[/latex]
• Arc Length of a Function of [latex]y[/latex]
  [latex]\text{Arc Length}={\displaystyle\int }_{c}^{d}\sqrt{1+{\left[{g}^{\prime }(y)\right]}^{2}}dy[/latex]
• Surface Area of a Function of [latex]x[/latex]
  [latex]\text{Surface Area}={\displaystyle\int }_{a}^{b}(2\pi f(x)\sqrt{1+{({f}^{\prime }(x))}^{2}})dx[/latex]

Glossary

arc length

the arc length of a curve can be thought of as the distance a person would travel along the path of the curve

cross-section

the intersection of a plane and a solid object

disk method

a special case of the slicing method used with solids of revolution when the slices are disks

frustum

a portion of a cone; a frustum is constructed by cutting the cone with a plane parallel to the base

method of cylindrical shells

a method of calculating the volume of a solid of revolution by dividing the solid into nested cylindrical shells; this method is different from the methods of disks or washers in that we integrate with respect to the opposite variable

slicing method

a method of calculating the volume of a solid that involves cutting the solid into pieces, estimating the volume of each piece, then adding these estimates to arrive at an estimate of the total volume; as the number of slices goes to infinity, this estimate becomes an integral that gives the exact value of the volume

solid of revolution

a solid generated by revolving a region in a plane around a line in that plane

surface area

the surface area of a solid is the total area of the outer layer of the object; for objects such as cubes or bricks, the surface area of the object is the sum of the areas of all of its faces

washer method

a special case of the slicing method used with solids of revolution when the slices are washers

Study Tips

Area of a Region between Two Curves

• Practice identifying which function is upper and which is lower in different scenarios.
• Remember that the area formula assumes [latex]f(x) ≥ g(x)[/latex]. If this isn’t true, swap the functions or split the interval.
• Sketch the region to visualize the problem and check your answer’s reasonableness.
• When finding intersection points, consider both algebraic and graphical methods.
• Pay attention to units, especially in applied problems.

Areas of Compound Regions

• Sketch the functions to visualize intersection points and subregions
• Remember: [latex]|f(x) - g(x)| = f(x) - g(x)[/latex] when [latex]f(x) \geq g(x)[/latex], and vice versa
• Be comfortable switching the order of functions within the absolute value signs
• Use trigonometric identities when working with trigonometric functions
• Check your work by ensuring areas are positive in each subinterval

Regions Defined with Respect to [latex]y[/latex]

• Practice rewriting functions of [latex]x[/latex] in terms of [latex]y[/latex]
• Sketch the region to visualize which function forms the right and left boundaries
• Pay attention to the domain of the new functions of [latex]y[/latex]
• Remember that the limits of integration are now [latex]y[/latex]-values, not [latex]x[/latex]-values
• Compare results with integration with respect to x as a check
• Consider which method ([latex]x[/latex] or [latex]y[/latex] integration) might be easier before starting
• Be aware of regions where integration with respect to [latex]y[/latex] might not be possible

Volume and the Slicing Method

• Visualize the solid and its cross-sections
• Remember that the area formula and integration limits depend on the chosen axis
• Be comfortable with variable cross-sectional areas
• Consider symmetry to simplify calculations

Solids of Revolution and the Slicing Method

• Practice sketching solids of revolution from 2D regions
• Remember that the radius of each circular cross-section is the distance from the function to the axis of rotation
• Be careful with the limits of integration – they should be [latex]x[/latex]-values, not [latex]y[/latex]-values

The Disk Method

• Visualize the solid of revolution to understand the shape of the disks
• Practice setting up integrals for rotation around both [latex]x[/latex] and [latex]y[/latex] axes
• Remember that the radius of each disk is the distance from the function to the axis of rotation
• Pay attention to the orientation of the disks (vertical for [latex]x[/latex]-axis rotation, horizontal for [latex]y[/latex]-axis)
• Be careful with the limits of integration – they should match the axis of rotation

The Washer Method

• Visualize the solid of revolution to understand the shape of the washers
• Practice identifying the outer and inner functions for different scenarios
• Remember that the washer thickness is [latex]dx[/latex] for [latex]x[/latex]-axis rotation and [latex]dy[/latex] for [latex]y[/latex]-axis rotation
• For rotation around lines other than axes, adjust the radii accordingly
• Be careful with the limits of integration – they should match the bounds of the region

Cylindrical Shells Method

• Practice visualizing the shells and how they form the solid
• Remember that shells are always parallel to the axis of rotation
• For [latex]y[/latex]-axis rotations, try cylindrical shells first if functions are defined in terms of [latex]x[/latex]
• Be comfortable adjusting the formula for different axes of rotation
• Compare setups using different methods to determine the easiest approach

Arc Lengths of Curves

• Practice visualizing the curve and its approximating line segments
• Remember to square the derivative inside the square root
• Be prepared to use u-substitution or other integration techniques

Area of a Surface of Revolution

• Visualize the surface of revolution and its approximating frustums
• Remember to square the derivative inside the square root
• Be prepared to use u-substitution or other integration techniques