Question

This question is based on the below graph: Which number below represents the area of the shaded curve to the closest value? 

• A. 1
• B. 1.5
• C. 2
• D. 3

Hint:

When exact integration is difficult, estimate the area using geometric shapes or numerical methods.

Answer 1

The Correct Option is C

1. Analyze the Graph: The graph shows a curve that starts at the origin (0,0), rises to a maximum height of 1 at approximately x = 1.5-1.6, and returns to 0 at approximately x = 3.1. This shape strongly resembles one arch of a sine wave.
2. Identify a Possible Function: A standard sine function, \(y = \sin(x)\), completes one arch from \(x=0\) to \(x=\pi\). Note that \(\pi \approx 3.14\), which matches the approximate endpoint on the graph's x-axis. The maximum value of \(\sin(x)\) is 1, which also matches the graph's peak height. Thus, it's highly probable that the curve represents \(y = \sin(x)\) over the interval \([0, \pi]\).
3. Calculate the Area using Integration: The area under the curve \(y = f(x)\) from \(x=a\) to \(x=b\) is given by the definite integral \(\int_a^b f(x) \, dx\). For \(y = \sin(x)\) from \(0\) to \(\pi\), the area is:

\[ \text{Area} = \int_{0}^{\pi} \sin(x) \, dx \]

The integral of \(\sin(x)\) is \(-\cos(x)\). Evaluating the definite integral:

\[ \text{Area} = [-\cos(x)]_{0}^{\pi} = (-\cos(\pi)) - (-\cos(0)) \]

We know that \(\cos(\pi) = -1\) and \(\cos(0) = 1\). Substituting these values:

\[ \text{Area} = (-(-1)) - (-1) = 1 - (-1) = 1 + 1 = 2 \]

1. Estimate the Area Visually: The shaded area is contained within a rectangle with width approximately \(\pi \approx 3.14\) and height 1. The area of this rectangle is about 3.14. The shaded area covers a significant portion of this rectangle, definitely more than half (which would be ~1.57). A triangle with the same base and height would have an area of \(\frac{1}{2} \times \pi \times 1 \approx 1.57\). Since the curve is above the triangle's straight lines, the area must be greater than 1.57. The value 2 seems like a plausible estimate.
2. Compare with Options: The calculated area is exactly 2. The options are (A) 1, (B) 1.5, (C) 2, (D) 3.

The calculated area matches option (C).

The closest value representing the area of the shaded curve is 2.

Answer 2

The yellow shaded region appears to be under a symmetric curve from \( x = 0 \) to \( x = 3 \), with a maximum height of \( y = 1 \). We approximate this region using a sine curve: 

Assume the curve is \( y = \sin\left(\frac{\pi x}{3}\right) \). The area under this curve from \( x = 0 \) to \( x = 3 \) is:

\[ \int_0^3 \sin\left(\frac{\pi x}{3}\right)\, dx = -\frac{3}{\pi} \cos\left(\frac{\pi x}{3}\right) \Big|_0^3 \] \[ = -\frac{3}{\pi} \left[\cos(\pi) - \cos(0)\right] = -\frac{3}{\pi} \left[-1 - 1\right] = \frac{6}{\pi} \approx 1.91 \]

Therefore, the area is approximately 1.91, which is closest to option (C) 2.