Question

61–62. Points of intersection and areab. 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

Accepted Answer

Identify the region bounded by the curves $$f(x) = \sinh x$$, $$g(x) = 	anh x$$, 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 $$x \u003c \ln 3. $$Set the two functions equal to find their points of intersection: $$\sinh x = 	anh x. $$Solve this equation to find the $$x-$$values where the curves meet. These will serve as the limits of integration along with $$x = \ln 3. $$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. Set up the definite integral for the area: $$	ext{Area} = \int_{a}^{\ln 3} \bigl| f(x) - g(x) \bigr| \, dx$$, 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 $$\int_{a}^{\ln 3} (	ext{upper function} - 	ext{lower function}) \, dx. $$Evaluate the integral by integrating the difference of the functions. Recall the derivatives and integrals of hyperbolic functions: $$\frac{d}{dx} \sinh x = \cosh x$$, $$\frac{d}{dx} 	anh x = 	ext{sech}^2 x. $$Use these to find antiderivatives and then apply the Fundamental Theorem of Calculus to compute the definite integral.

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

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Key Concepts

Hyperbolic Functions

Finding Points of Intersection

Definite Integral for Area Between Curves