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Magazine Chapter 6: Problem 87
Sketch the parabola \(y=x(x-\pi)\) and the curve \(y=\) \(\sin x,\) showing their points of intersection. Find the exact area between the two graphs.
Short Answer
Expert verified
The area is \(2 + \frac{\pi^3}{6}\).
Step by step solution
01
Identify the Intersections
First, we need to find the points of intersection of the parabola \(y = x(x-\pi)\) and the sine curve \(y = \sin x\). To do this, set them equal to each other: \[x(x-\pi) = \sin x.\] We need to solve this equation for \(x\). Observing the functions, the likely intersections are at \(x=0\) and \(x=\pi\). Compare the values: - For \(x = 0\), \(0(0-\pi) = \sin 0 = 0\), which matches. - For \(x = \pi\), \(\pi(0) = \sin \pi = 0\), which also matches. Thus, the points of intersection are at \((0, 0)\) and \((\pi, 0)\).
02
Setup the Integral for Area
The area between two curves from \(x = a\) to \(x = b\) is given by the integral: \[\int_{a}^{b} (\text{top function} - \text{bottom function}) \, dx.\]In our case, from \(x = 0\) to \(x = \pi\), the sine function \(y = \sin x\) is on top (since \(\sin x\) is non-negative in this interval and the parabola is zero at the endpoints), and the parabola \(y = x(x-\pi)\) is below it. Therefore, set up the integral as: \[\int_{0}^{\pi} (\sin x - x(x-\pi)) \, dx.\]
03
Simplify the Integration Setup
Simplify the expression inside the integral: \[\sin x - x(x-\pi) = \sin x - (x^2 - \pi x) = \sin x + \pi x - x^2.\]Thus, the integral becomes:\[\int_{0}^{\pi} (\sin x + \pi x - x^2) \, dx.\]
04
Integrate the Function
Now, integrate each term separately. The integral is:\[\int \sin x \, dx = -\cos x \]\[\int \pi x \, dx = \frac{\pi}{2}x^2 \] \[\int x^2 \, dx = \frac{x^3}{3}.\]Thus, the definite integral becomes:\[\left[ -\cos x + \frac{\pi}{2}x^2 - \frac{x^3}{3} \right]_{0}^{\pi}.\]
05
Evaluate the Definite Integral
Substitute the bounds and evaluate:Evaluate at \(x = \pi\):\[-\cos \pi + \frac{\pi}{2} \pi^2 - \frac{\pi^3}{3} = 1 + \frac{\pi^3}{2} - \frac{\pi^3}{3}.\]Evaluate at \(x = 0\):\[-\cos 0 + 0 - 0 = -1.\]Find the difference:\[(1 + \frac{\pi^3}{2} - \frac{\pi^3}{3}) - (-1) = 2 + \frac{\pi^3}{2} - \frac{\pi^3}{3}.\]
06
Compute the Exact Area
Simplify the expression for the area:\[2 + \frac{\pi^3}{2} - \frac{\pi^3}{3} = 2 + \frac{3\pi^3}{6} - \frac{2\pi^3}{6}.\]Combine the terms:\[2 + \frac{\pi^3}{6}.\]
07
Conclusion
The exact area between the parabola \(y = x(x - \pi)\) and the sine curve \(y = \sin x\) from \(x = 0\) to \(x = \pi\) is \(2 + \frac{\pi^3}{6}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabola
To understand the problem, imagine a shape that looks like a shallow "U" or "V." This shape is a parabola. The one relevant here is defined by the equation \( y = x(x - \pi) \). This equation forms a downward-opening parabola, meaning it curves towards the bottom of the graph. Some key features of a parabola include:
- The vertex is its highest or lowest point. For \( y = x(x - \pi) \), it doesn't pass through the origin symmetrically about the y-axis, but still has points of intersection.
- The axis of symmetry helps in understanding its structure and solving for intersection points with other curves like a sine function.
Sine Function
The sine function is one of the basic trigonometric functions, and it creates a wave-like pattern. When graphed, it smoothly oscillates between -1 and 1. For the equation \( y = \sin x \), it's essential to know:
- The sine wave starts at (0,0), creeps up to 1, and progresses to negative values within intervals between 0 and \( \pi \).
- Because the sine wave oscillates smoothly and predictably, it often forms interesting shapes when plotted with other functions, like parabolas.
Definite Integral
The definite integral is a powerful tool in calculus used to calculate the area under a curve. Here, it helps us find the specific area between two intersecting curves over a given interval. To set up a definite integral, consider:
- The function defined on an interval \([a, b]\), where \(a\) and \(b\) are the limits of integration.
- The process of integration, which at its core, sums up infinite infinitesimally small elements to compute a cumulative area.
Area Between Curves
Determining the area between two functions involves understanding their positions relative to each other across a specific interval. In this exercise, the objective was to find the area between the parabola \( y = x(x-\pi) \) and the sine function \( y = \sin x \) from \(x = 0\) to \(x = \pi\). Here鈥檚 how it works:
- Identify which function is "on top" and which one is "below" across the interval. This ensures accurate area calculation.
- Set up the problem as a definite integral, integrating the difference between the top and bottom functions over the chosen interval.
