Understanding the Area of a Loop for y = c sin x

The question asks for the area of one of the loops formed between the curve \(y = c \sin x\) and the x-axis. The curve \(y = c \sin x\) is a sinusoidal function, which oscillates around the x-axis. The "loops" are the regions enclosed by the curve and the x-axis.

Finding the X-Intercepts

The curve intersects the x-axis when \(y = 0\). So, we set \(c \sin x = 0\).

Assuming \(c \neq 0\), this occurs when \(\sin x = 0\).

The general solution for \(\sin x = 0\) is \(x = n\pi\), where \(n\) is an integer (\(n \in \mathbb{Z}\)).

The consecutive x-intercepts are at \(x = 0, \pi, 2\pi, 3\pi, \ldots\) and \(x = -\pi, -2\pi, -3\pi, \ldots\). Each interval between two consecutive intercepts on the x-axis defines a loop.

Identifying One Loop

Let's consider the first loop formed for \(x \ge 0\). The first two consecutive intercepts are at \(x = 0\) and \(x = \pi\). The region between the curve \(y = c \sin x\) and the x-axis for \(x\) from \(0\) to \(\pi\) forms one loop.

For \(x \in (0, \pi)\), the value of \(\sin x\) is positive. If \(c > 0\), then \(y = c \sin x\) is positive in this interval, and the loop is above the x-axis. If \(c < 0\), then \(y = c \sin x\) is negative in this interval, and the loop is below the x-axis.

The area of a region between a curve \(y = f(x)\) and the x-axis from \(x=a\) to \(x=b\) is given by the definite integral \(\int_a^b |f(x)| dx\). To find the area of one loop, we integrate from \(x=0\) to \(x=\pi\):

\(\text{Area} = \int_0^\pi |c \sin x| dx\)

Calculating the Area

Let's evaluate the integral. The absolute value \(|c \sin x|\) depends on the sign of \(c\).

If \(c \ge 0\), then \(|c \sin x| = c \sin x\) for \(x \in [0, \pi]\) because \(\sin x \ge 0\) in this interval.

If \(c < 0\), then \(|c \sin x| = -c \sin x\) for \(x \in [0, \pi]\) because \(\sin x \ge 0\) and \(c\) is negative, making \(c \sin x \le 0\).

In either case, the integral involves integrating \(c \sin x\) or \(-c \sin x\), and the constant factor \(c\) or \(-c\) comes out of the integral.

Let's assume \(c > 0\) for simplicity, as the options suggest a result in terms of \(c\) with a positive coefficient.

\(\text{Area} = \int_0^\pi c \sin x dx\)

We can pull the constant \(c\) out of the integral:

\(\text{Area} = c \int_0^\pi \sin x dx\)

The antiderivative of \(\sin x\) is \(-\cos x\). Now we evaluate the definite integral:

\(\text{Area} = c [-\cos x]_0^\pi\)

Substitute the limits of integration:

\(\text{Area} = c (-\cos(\pi) - (-\cos(0)))\)

\(\text{Area} = c (-(-1) - (-1))\)

\(\text{Area} = c (1 + 1)\)

\(\text{Area} = 2c\)

If \(c < 0\), the area would be \(\int_0^\pi -c \sin x dx = -c \int_0^\pi \sin x dx = -c[-\cos x]_0^\pi = -c(1+1) = -2c\). Since \(c\) is negative, \(-2c\) is a positive value, representing the area. The options are given in terms of \(c\), and \(2c\) is listed. This implies that either \(c\) is assumed to be positive or the options list the magnitude \(|2c|\) in terms of \(c\).

The calculated area of one loop between \(x=0\) and \(x=\pi\) is \(2c\) (assuming \(c>0\) or representing the magnitude). The loops between \(\pi\) and \(2\pi\), \(2\pi\) and \(3\pi\), etc., will have the same area magnitude.

Summary of Steps

Identify the x-intercepts by setting \(y = 0\).

Choose an interval between two consecutive intercepts that defines one loop (e.g., \(0\) to \(\pi\)).

Set up the definite integral of the absolute value of the function over this interval to find the area: \(\int_0^\pi |c \sin x| dx\).

Evaluate the integral. Assuming \(c > 0\), \(\int_0^\pi c \sin x dx = 2c\). If \(c < 0\), the area magnitude is \(-2c\), giving a positive value.

The result of the integration for one loop between the curve \(y = c \sin x\) and the x-axis is \(2c\) (assuming \(c>0\)).

Revision Table: Area Under Curve Concepts

Additional Information: Properties of Sine Function

The sine function, \(y = \sin x\), is a periodic function with a period of \(2\pi\). The curve oscillates between -1 and 1. The curve \(y = c \sin x\) scales this vertical oscillation by a factor of \(c\).

The graph of \(y = c \sin x\) crosses the x-axis at multiples of \(\pi\) (\(0, \pm\pi, \pm2\pi, \ldots\)).

Between \(0\) and \(\pi\), \(\sin x \ge 0\). The loop is above the x-axis if \(c > 0\) and below if \(c < 0\).

Between \(\pi\) and \(2\pi\), \(\sin x \le 0\). The loop is below the x-axis if \(c > 0\) and above if \(c < 0\).

The shape and area of each successive loop (between \(n\pi\) and \((n+1)\pi\)) are the same in magnitude.

Understanding the graph of \(y = c \sin x\) helps visualize the loops whose areas are being calculated.