6.1 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 y to find the area. The principles are the same regardless of which variable is used as the variable of integration.

6.2 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.

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.

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 $y .$
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.

Several physical applications of the definite integral are common in engineering and physics.
Definite integrals can be used to determine the mass of an object if its density function is known.
Work can also be calculated from integrating a force function, or when counteracting the force of gravity, as in a pumping problem.
Definite integrals can also be used to calculate the force exerted on an object submerged in a liquid.

Mathematically, the center of mass of a system is the point at which the total mass of the system could be concentrated without changing the moment. Loosely speaking, the center of mass can be thought of as the balancing point of the system.
For point masses distributed along a number line, the moment of the system with respect to the origin is $M = \sum_{i = 1}^{n} m_{i} x_{i} .$ For point masses distributed in a plane, the moments of the system with respect to the x- and y-axes, respectively, are $M_{x} = \sum_{i = 1}^{n} m_{i} y_{i}$ and $M_{y} = \sum_{i = 1}^{n} m_{i} x_{i},$ respectively.
For a lamina bounded above by a function $f (x),$ the moments of the system with respect to the x- and y-axes, respectively, are $M_{x} = ρ \int_{a}^{b} \frac{{[f (x)]}^{2}}{2} d x$ and $M_{y} = ρ \int_{a}^{b} x f (x) d x .$
The x- and y-coordinates of the center of mass can be found by dividing the moments around the y-axis and around the x-axis, respectively, by the total mass. The symmetry principle says that if a region is symmetric with respect to a line, then the centroid of the region lies on the line.
The theorem of Pappus for volume says that if a region is revolved around an external axis, the volume of the resulting solid is equal to the area of the region multiplied by the distance traveled by the centroid of the region.

The earlier treatment of logarithms and exponential functions did not define the functions precisely and formally. This section develops the concepts in a mathematically rigorous way.
The cornerstone of the development is the definition of the natural logarithm in terms of an integral.
The function $e^{x}$ is then defined as the inverse of the natural logarithm.
General exponential functions are defined in terms of $e^{x},$ and the corresponding inverse functions are general logarithms.
Familiar properties of logarithms and exponents still hold in this more rigorous context.

Exponential growth and exponential decay are two of the most common applications of exponential functions.
Systems that exhibit exponential growth follow a model of the form $y = y_{0} e^{k t} .$
In exponential growth, the rate of growth is proportional to the quantity present. In other words, $y^{'} = k y .$
Systems that exhibit exponential growth have a constant doubling time, which is given by $(ln 2) / k .$
Systems that exhibit exponential decay follow a model of the form $y = y_{0} e^{− k t} .$
Systems that exhibit exponential decay have a constant half-life, which is given by $(ln 2) / k .$

Hyperbolic functions are defined in terms of exponential functions.
Term-by-term differentiation yields differentiation formulas for the hyperbolic functions. These differentiation formulas give rise, in turn, to integration formulas.
With appropriate range restrictions, the hyperbolic functions all have inverses.
Implicit differentiation yields differentiation formulas for the inverse hyperbolic functions, which in turn give rise to integration formulas.
The most common physical applications of hyperbolic functions are calculations involving catenaries.