Question

The area of the region bounded by curves \(f(x)=\sin x\) \(g(x)=\cos x, x=(\pi / 4), x=(5 \pi / 4)\) is........ (a) 1 (b) 2 (c) \(\sqrt{2}\) (d) \(2 \sqrt{2}\)

Short answer

The area of the region bounded by the curves \(f(x)=\sin x\), \(g(x)=\cos x\), \(x=(\pi / 4)\), and \(x=(5 \pi / 4)\) is (d) \(2 \sqrt{2}\).

Step-by-step solution

Determine which function is greater in the given interval

First, let's find where the two functions are equal by solving the following equation: \(f(x)=g(x)\) \[ \sin x = \cos x \] We can rewrite this equation as: \[ \tan x = 1 \] We know that \(\tan x\) will be equal to 1 at \(x = \pi/4\) (and the series \((2n + 1)\pi/4\), where \(n\) is an integer). Now let's check the given interval. Since \(x = (\pi/4)\) is one of the endpoints and \(\tan x = 1\) at that point, we need to check within the interval. For \(x=(\pi/2)\), which falls between the given endpoints, we get \(f(x) = \sin(\pi/2) = 1\) and \(g(x) = \cos(\pi/2) = 0\). Therefore, \(f(x) > g(x)\) within the interval.

Integrate the difference within the interval

Since \(f(x) > g(x)\) in the interval, we will integrate the difference \((\sin x - \cos x)\) between the two functions. Let \(A\) be the area of the region: \[ A = \int_{\pi/4}^{5\pi/4} (\sin x - \cos x) dx \]

Integrate and substitute the limits of integration

Now integrate the functions separately and evaluate the definite integral: \[ A = \left[-\cos x - \sin x\right]_{\pi/4}^{5\pi/4} \] Now, substitute the limits of integration: \[ A = \left[(-\cos(5\pi/4)-\sin(5\pi/4))\) - \((-cos(\pi/4)-\sin(\pi/4))\right] \] By simplifying the trigonometric expressions, we get: \[ A = [(\sqrt{2}/2-\sqrt{2}/2)]-[(-\sqrt{2}/2+\sqrt{2}/2)] \] \[ A = 2(\sqrt{2}/2 - (-\sqrt{2}/2)) \] \[ A = 2(\sqrt{2}) \] Therefore, the answer is (d) \(2\sqrt{2}\).

Key concepts

Definite Integration

Definite integration is a key concept in calculus, allowing us to calculate the area under a curve within a specified interval. In simpler terms, it tells us how much space a certain part of a graph covers along the x-axis. For instance, if we consider the functions \(f(x) = \sin x\) and \(g(x) = \cos x\), and we want to find the area between these curves from \(x = \pi/4\) to \(x = 5\pi/4\), we apply definite integration.

We calculate the integral of the difference \((\sin x - \cos x)\) across this interval. This process, known as integrating with limits, helps us accumulate the difference in height between the two curves, giving us the area between them. The integration step involves:

• Identifying the upper function and the lower function within the interval.
• Subtracting the lower function from the upper function.
• Evaluating the integral from the specified lower limit to the upper limit.

When you substitute the upper and lower limits into the integrated function, you obtain a concrete area value between two curves.

Trigonometric Functions

Trigonometric functions such as sine and cosine are periodic and oscillate between fixed values. These properties make them interesting to study, especially when graphing their behavior. In this exercise, we work with \(f(x) = \sin x\) and \(g(x) = \cos x\), both classic trigonometric functions. The sine function starts at zero, rises to one, falls to zero, plunges to negative one, and then returns to zero over a range of \(0\) to \(2\pi\). The cosine function follows a similar path but starts at one, falls to zero, down to negative one, and back to one.

Understanding the comparative behavior of these functions is essential to identify which is greater at different points along the x-axis. For the interval \(\pi/4\) to \(5\pi/4\), knowing \(\sin x > \cos x\) helps determine that \(\sin x\) will be the upper curve in this case. Using this knowledge simplifies calculating the area between these functions using definite integration.

Limits of Integration

Limits of integration define the start and end points for evaluating an integral and play a crucial role in determining the area under a curve. In our example, these limits are \(x = \pi/4\) and \(x = 5\pi/4\). These points help confine the area calculation to a manageable portion of the graph where we can compare the behavior of \(\sin x\) and \(\cos x\).

When we integrate the function \(\sin x - \cos x\), the limits \(\pi/4\) and \(5\pi/4\) guide us in evaluating the antiderivatives at specific intervals. By substituting these endpoints into the integrated expression, we determine the net accumulated value, representing the total area between the curves over this span. This difference, effectively calculated, forms the crux of evaluating definite integrals, turning abstract functions into tangible, visualizable areas.