Problem 81 - Calculus Early Transcendentals

Question

Find the area of the region bounded by \(y=\operatorname{sech} x, x=1,\) and the unit circle.

Step-by-Step Solution

Answer

Answer: The approximate area of the region is 1.215 square units.

Step 1: Find the intersection points with the unit circle

First, let's find the intersection points between \(y=\operatorname{sech} x\) and the unit circle. The equation of the unit circle is \(x^2 + y^2 =1\). We can solve for y in terms of x, which gives us \(y=\sqrt{1-x^2}\). Now, we have to find x such that \(\sqrt{1-x^2}=\operatorname{sech} x\): \(\sqrt{1-x^2}=\operatorname{sech} x\) Since the hyperbolic secant function is always positive, we only need to consider the positive square root because our region lies above the x-axis. Now, square both sides of the equation to remove the square root: \(1-x^2=\operatorname{sech}^2 x\) Recall that \(\operatorname{sech}^2 x = 1 -\operatorname{tanh}^2 x\). Substituting this into the equation, we get: \(1-x^2=1-\operatorname{tanh}^2 x\) Since \(x=1\) is given as one of the bounds, we just need to find the intersection point with the unit circle on the left side. We can do this by solving the equation for x: \(x^2=\operatorname{tanh}^2 x\) But notice that \(\operatorname{tanh} x = \dfrac{e^x-e^{-x}}{e^x+e^{-x}}\), so the equation won't have an explicit solution for x. As a result, we need to find the intersection numerically. Using a calculator or a numerical method like Newton's method, we find that the intersection occurs at \(x \approx -0.6\) We can then find the y-value at this x-coordinate: \(y = \operatorname{sech} (-0.6) \approx 0.825\) So, our region is bounded by the points \((-0.6,0.825)\) and \((1,0)\).

Step 2: Set up the integral to find the area

To find the area of the region bounded by the two curves, we'll use the following formula for the area enclosed between two curves: \(A=\int_a^b \Big|f(x)-g(x)\Big| dx\) where f(x) and g(x) are the functions of the two curves, and [a,b] are the x-limits of integration. In our case, the limits of integration are [-0.6, 1], and our functions are \(f(x) = \operatorname{sech} x\) and \(g(x) = \sqrt{1-x^2}\), so our integral becomes: \(A=\int_{-0.6}^1 \Big|\operatorname{sech} x-\sqrt{1-x^2}\Big| dx\) Since the region is entirely above the x-axis, we know that both functions are positive in the given interval, and we can remove the absolute value signs: \(A=\int_{-0.6}^1 \Big(\operatorname{sech} x-\sqrt{1-x^2}\Big) dx\)

Step 3: Evaluate the integral and find the area

Unfortunately, the integral has no closed-form expression involving elementary functions. However, we can use numerical integration methods, such as the trapezoidal rule or Simpson's rule, or a calculator with a numerical integration function to find an approximate value for the integral: \(A\approx 1.215\) Thus, the area of the region bounded by \(y=\operatorname{sech} x\), \(x=1\), and the unit circle is approximately 1.215 square units.