61–62. Points of intersection and area
b. Compute the area of the region described.


f(x) = sinh x, g(x) = tanh x; the region bounded by the graphs of f, g, and x = ln 3

Verified step by step guidance

1. Identify the region bounded by the curves \(f(x) = \sinh x\), \(g(x) = \tanh x\), and the vertical line \(x = \ln 3\). To find the area, we need to determine the interval over which to integrate. Since the problem specifies \(x = \ln 3\) as a boundary, find the other boundary by finding the point where \(f(x)\) and \(g(x)\) intersect for \(x < \ln 3\).
2. Set the two functions equal to find their points of intersection: \(\sinh x = \tanh x\). Solve this equation to find the \(x\)-values where the curves meet. These will serve as the limits of integration along with \(x = \ln 3\).
3. Determine which function is on top and which is on the bottom between the limits of integration. This is important because the area between two curves \(f(x)\) and \(g(x)\) from \(a\) to \(b\) is given by \(\int_a^b |f(x) - g(x)| \, dx\). Identify \(f(x)\) or \(g(x)\) as the upper function and the other as the lower function in this interval.
4. Set up the definite integral for the area: \(\text{Area} = \int_{a}^{\ln 3} \bigl| f(x) - g(x) \bigr| \, dx\), where \(a\) is the intersection point found in step 2. Since you know which function is on top, you can write the integral without absolute value as \(\int_{a}^{\ln 3} (\text{upper function} - \text{lower function}) \, dx\).
5. Evaluate the integral by integrating the difference of the functions. Recall the derivatives and integrals of hyperbolic functions: \(\frac{d}{dx} \sinh x = \cosh x\), \(\frac{d}{dx} \tanh x = \text{sech}^2 x\). Use these to find antiderivatives and then apply the Fundamental Theorem of Calculus to compute the definite integral.

Key Concepts

Hyperbolic Functions

Hyperbolic functions like sinh(x) and tanh(x) are analogs of trigonometric functions but based on hyperbolas. sinh(x) = (e^x - e^{-x})/2 and tanh(x) = sinh(x)/cosh(x). Understanding their properties and graphs is essential to identify the region bounded by these curves.

Finding Points of Intersection

To determine the bounded region, it is crucial to find where the two functions intersect. This involves solving the equation sinh(x) = tanh(x) to find the x-values where the curves meet, which define the limits of integration for the area calculation.

Definite Integral for Area Between Curves

The area between two curves f(x) and g(x) from a to b is found by integrating the difference |f(x) - g(x)| dx over [a, b]. Identifying which function is on top in the interval and using the given boundary x = ln(3) allows computation of the enclosed area.