Problem 40 - Calculus Early Transcendentals

Question

Net area from graphs The accompanying figure shows four regions bounded by the graph of \(y=x \sin x: R_{1}, R_{2}, R_{3},\) and \(R_{4},\) whose areas are \(1, \pi-1, \pi+1,\) and \(2 \pi-1,\) respectively. (We verify these results later in the text.) Use this information to evaluate the following integrals. $$\int_{\pi / 2}^{2 \pi} x \sin x d x$$

Step-by-Step Solution

Answer

Question: Evaluate the integral \(\int_{\pi / 2}^{2 \pi} x \sin x d x\), provided that the areas of regions \(R_1 = 1, R_2 = \pi - 1, R_3 = \pi + 1\), and \(R_4\) is outside the given interval. Answer: \(-1\)

Step 1: Identify the Boundaries of Regions R1, R2, R3, and R4

We can find the boundaries of regions \(R_1, R_2, R_3\), and \(R_4\) by identifying where the graph of \(y = x \sin x\) intersects the x-axis (i.e., where \(\sin x = 0\)). These intersection points occur at \((n\pi, 0)\) for every integer \(n\). We know that the regions \(R_1, R_2, R_3\), and \(R_4\) are all bounded by the curve and x-axis, so we need to find the x-values where these regions start and end. As \(\int_{\pi / 2}^{2 \pi} x \sin x d x\) is considered, we have regions starting from \(\pi/2\) until \(2\pi\). Thus, we have the following: - \(R_1\) starts at \(\pi/2\) and ends at \(\pi\); - \(R_2\) starts at \(\pi\) and ends at \(3\pi/2\); - \(R_3\) starts at \(3\pi/2\) and ends at \(2\pi\); - \(R_4\) is outside of the given integral interval, so we can ignore it.

Step 2: Split the Integral into Sub-Integrals

Since we have the boundaries of the regions, we can split the given integral into three sub-integrals as below: $$\int_{\pi / 2}^{2 \pi} x \sin x d x = \int_{\pi / 2}^{\pi} x \sin x d x + \int_{\pi}^{3 \pi / 2} x \sin x d x + \int_{3 \pi / 2}^{2 \pi} x \sin x d x$$

Step 3: Determine the Sign of Each Sub-Integral

The integral \(\int x\sin x dx\) will have positive values if the integrand \(x\sin x\) is positive and negative values if the integrand is negative. From the boundaries we derived in step 1, we know: - In the interval \([\pi / 2, \pi]\), \(x\) is positive, and \(\sin x > 0\), making the integrand positive in \(R_1\); - In the interval \([\pi, 3 \pi / 2]\), both \(x\) and \(\sin x\) are positive, so the integrand is positive in \(R_2\); - In the interval \([3 \pi / 2, 2 \pi]\), \(x\) is positive, but \(\sin x < 0\), making the integrand negative in \(R_3\).

Step 4: Calculate the Net Area Using the Given Information

Using our findings in Step 3, we can calculate the net area of the integral as follows: $$\int_{\pi / 2}^{2 \pi} x \sin x d x = \int_{\pi / 2}^{\pi} x \sin x d x - \int_{3 \pi / 2}^{2 \pi} x \sin x d x$$ With given values for the areas of regions \(R_1 = 1\), \(R_2 = \pi - 1\), and \(R_3 = \pi + 1\), we compute: $$\int_{\pi / 2}^{2 \pi} x \sin x d x = R_1 - (R_3 - R_2) = 1 - (\pi + 1 - (\pi - 1)) = 1 - 2 = -1$$ Therefore, the value of the given integral is: $$\int_{\pi / 2}^{2 \pi} x \sin x d x = -1$$.

Key Concepts

Integration by Parts

Integration by parts is a powerful technique used to solve integrals of products of functions, and it is based on the product rule for differentiation. The formula for integration by parts is \[ \int u dv = uv - \int v du \.\] To apply this technique, we first identify parts of the integrand that can be set as \(u\) (which will be differentiated) and \(dv\) (which will be integrated). The strategy involves choosing \(u\) to be a function that simplifies when differentiated and \(dv\) such that it is relatively easy to integrate. ewline After applying the formula, we often end up with a simpler integral that is easier to solve. In our exercise, integration by parts could be used, but the problem simplifies by considering the geometric interpretation of the definite integral as an area under the curve.

Area Under a Curve

The definite integral of a function between two limits has a geometric interpretation as the net area bounded by the function's graph, the x-axis, and the vertical lines corresponding to the limits of integration. This concept enables us to visualize integration as the process of adding up infinitely many infinitesimally thin rectangles under the curve.ewline When the graph of the function lies above the x-axis, the area is considered positive, and when it lies below, the area is negative. Calculating the definite integral, therefore, gives us the net area which is the total area above the x-axis minus the total area below the x-axis within the given interval.

Sine Function Properties

The sine function exhibits certain properties that are highly relevant when working with integrals involving trigonometric functions. One property is that sine is a periodic function with a period of \(2\pi\), which means the function repeats its values every \(2\pi\) units along the x-axis. Additionally, the sine function is symmetrical about the origin, implying that \(\sin(-x) = -\sin(x)\), which reflects its odd function characteristic.ewline Furthermore, \(\sin x\) is positive in the first and second quadrants (\(0 < x < \pi\)) and negative in the third and fourth quadrants (\(\pi < x < 2\pi\)). This sign change affects the integral's evaluation, especially when assessing the net area under the curve. These properties help us understand the behavior of the sine function within the intervals and assist in breaking down the problem into manageable parts.

Net Area Calculation

Net area calculation involves determining the 'signed' area between the graph of a function and the x-axis over a certain interval. When computing the net area, we need to account for the fact that areas under the x-axis (where the function is negative) are subtracted from the areas above. ewline To find the net area for our exercise, we look at each interval's contribution to the integral and combine these values, accounting for their sign. This method is particularly useful when the function crosses the x-axis multiple times within the interval of integration. By summing these signed areas, we obtain the total net area under the curve from the start to the end of the interval. In the provided problem, we are evaluating the integral of \(x \sin x\) from \(\pi/2\) to \(2\pi\), which involves calculating the signed areas of three distinct regions and then combining them.