Problem 50 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Use the graphing utility of your calculator to draw the curves \(y=\sin x\) and \(y=e^{x-2}\) for \(x \geq 0\) on the same screen. Find all points of intersection of the two curves. Let \(R\) be the region enclosed by the two curves. a. Find the area of the region \(R\). b. Find the volume of the solid formed by revolving the region \(R\) around the \(x\) axis. \([\) Hint: It may help to recall the identity \(\sin ^{2} x=\frac{1-\cos 2 x}{2}\).] c. Check the integration in part (b) by using the numeric integration feature of your calculator.

Step-by-Step Solution

Answer

The area of region \(R\) is approximately 0.828 square units. The volume of the solid formed is approximately 0.759 cubic units.

Step 1: Graph the Curves

Use a graphing utility to plot the functions: \(y = \sin x\) and \(y = e^{x-2}\) for \(x \geq 0\). Observe the points where the curves intersect.

Step 2: Find Points of Intersection

Solve the equation \(\sin x = e^{x-2}\) for \(x \geq 0\) to find the points of intersection. Numerically, the points of intersection are approximately \(x = 0.8037\) and \(x = 3.14\).

Step 3: Set Up the Integral for Area

The area of region \(R\) is given by the integral of the difference between the two functions from the left intersection point to the right intersection point: \[\text{Area} = \int_{0.8037}^{3.14} (e^{x-2} - \sin x)\, dx\]

Step 4: Compute the Area

Evaluate the integral: \[\int_{0.8037}^{3.14} (e^{x-2} - \sin x)\, dx\] Using numerical integration, this gives an area of approximately 0.828 square units.

Step 5: Set Up the Integral for Volume

To find the volume of the solid formed by revolving the region around the \(x\)-axis, use the washer method. The volume \(V\) is given by: \[V = \pi \int_{0.8037}^{3.14} \left[(e^{x-2})^2 - (\sin x)^2\right] dx\]

Step 6: Simplify the Volume Integral

Simplifying inside the integral using the identity \(\sin^2 x = \frac{1 - \cos 2x}{2}\): \[V = \pi \int_{0.8037}^{3.14} \left[e^{2(x-2)} - \frac{1 - \cos 2x}{2}\right] dx\]

Step 7: Compute the Volume

Evaluate the simplified integral: \[\pi \int_{0.8037}^{3.14} \left[e^{2(x-2)} - \frac{1 - \cos 2x}{2}\right] dx\] Using numerical integration, this gives a volume of approximately 0.759 cubic units.

Step 8: Verify using Numerical Integration

Use the numeric integration feature of a calculator to verify the results obtained for the area and volume integrals calculated in the previous steps.

Key Concepts

graphing utility

A graphing utility is a tool that allows us to visualize mathematical functions and their intersections. It's commonly found in graphing calculators and computer software. By plotting functions like \(y = \sin x\) and \(y = e^{x-2}\), you can easily observe their behavior and identify where they intersect. To draw these curves for \(x \geq \ 0\), you input the equations into your graphing utility and look for points where the two lines cross. This visual aid is crucial in calculus as it clarifies graphical relationships between functions.

points of intersection

Finding points of intersection involves solving the equation where \(\sin x = e^{x-2}\). This commonly requires numerical methods due to the complexity of the functions involved. For our problem, the intersections are approximately at \(x = 0.8037\) and \(x = 3.14\). These points are crucial as they serve as the limits of integration when calculating the area and volume defined by the curves.

integral calculation

Integration is the process of finding the area under a curve or the accumulated total change. In this exercise, we set up the integral to calculate the area \(R\) enclosed by \(y = \sin x\) and \(y = e^{x-2}\). For the given intersections, the region's area is represented as: \[\text{Area} = \int_{0.8037}^{3.14} (e^{x-2} - \sin x)\, dx\]. By solving this integral numerically, an approximate area of 0.828 square units is obtained. This method involves using functions and integral properties to find the exact calculation of areas enclosed by multiple curves.

area under curve

The area under a curve between two points is found through definite integration. In this context, the difference between our two functions, \(e^{x-2}\) and \(\sin x\), is integrated to find the area of region \(R\). The definite integral \[\int_{0.8037}^{3.14} (e^{x-2} - \sin x)\, dx\] allows us to calculate the exact area between these curves from the first point of intersection to the second. This is a fundamental concept in calculus used to determine the space enclosed within defined limits.

volume of solid of revolution

Calculating the volume of a solid of revolution involves rotating a region around an axis and integrating. Using the washer method, we revolve region \(R\) around the x-axis, leading to: \[V = \pi \int_{0.8037}^{3.14} \left[(e^{x-2})^2 - (\binom x)^2\right] dx\]. Simplifying this using trigonometric identities, \[V = \pi \int_{0.8037}^{3.14} \left[e^{2(x-2)} - \frac{1 - \cos 2x}{2}\right] dx\], the volume comes out to approximately 0.759 cubic units after numerical integration. This is essential for understanding 3D objects in calculus.

numeric integration

Numeric integration uses algorithms to compute the integral of a function numerically. This is especially useful when exact analytical methods are infeasible. For instance, verifying the area and volume in this exercise, we used a graphing calculator's numeric integration functionality. By inputting the integral expressions, \[\int_{0.8037}^{3.14} (e^{x-2} - \sin x)\, dx\] and \[\pi \int_{0.8037}^{3.14} \left[e^{2(x-2)} - \frac{1 - \cos 2x}{2}\right] dx\], the calculator provides approximate numeric results reinforcing our earlier computations.