Problem 51 - Calculus: Early Transcendentals

Question

Use a graph to give a rough estimate of the area of the region that lies beneath the given curve. Then find the exact area. \( y = \sin x \), \( 0 \le x \le \pi \)

Step-by-Step Solution

Answer

The exact area under the curve is 2.

Step 1: Sketch the Graph of the Curve

First, sketch the graph of the function \( y = \sin x \) over the interval \( [0, \pi] \). This curve starts at the origin \( (0,0) \), reaches a maximum of \( y = 1 \) at \( x = \frac{\pi}{2} \), and returns to \( y = 0 \) at \( x = \pi \).

Step 2: Estimate the Area Using the Graph

Visualize the area under the curve from \( x = 0 \) to \( x = \pi \). The shape is reminiscent of half a circle with radius 1, so the area can be roughly estimated as about half the area of a circle with radius 1, which is \( \frac{1}{2} \pi \cdot 1^2 = \frac{\pi}{2} \).

Step 3: Set Up the Integral for Exact Area

To find the exact area under the curve, set up the definite integral of \( y = \sin x \) from \( x = 0 \) to \( x = \pi \). This integral is \( \int_0^{\pi} \sin x \, dx \).

Step 4: Solve the Integral

Solve the integral by finding the antiderivative of \( \sin x \), which is \( -\cos x \). Evaluate from \( 0 \) to \( \pi \):\[\int_0^{\pi} \sin x \, dx = \left[ -\cos x \right]_0^{\pi} = -\cos(\pi) + \cos(0) = -(-1) + 1 = 2.\]

Step 5: Confirm the Exact Area

The exact area under the curve \( y = \sin x \) from \( 0 \) to \( \pi \) is 2. Confirm that this value is consistent with the graph and integral results.

Key Concepts

Estimating Area Under a Curve

Understanding how to estimate the area under a curve is a fundamental skill in calculus. This process is crucial when trying to determine values for shapes that don’t have straightforward geometric formulas. Visual estimation involves looking at the curve and picturing simple shapes that can approximate the area. For instance, when dealing with a sine wave like \( y = \sin x \), the section from \( 0 \) to \( \pi \) forms a shape that resembles half of a circular arc.

Here are some points to consider when estimating areas:

• Assess the overall shape and symmetry: The section under the sine curve between \( 0 \) and \( \pi \) can be likened to a semi-circle because it follows a smooth, symmetrical arch.
• Use basic geometric formulas: You can use the formula for the area of a circle, \( \pi r^2 \), to find the area of semi-circular shapes by halving the result, as we did here with \( \frac{\pi}{2} \).
• Check for conciseness: Remember to always consider the boundaries. The limits \( 0 \le x \le \pi \) define where the curve starts and stops.

Visual estimation is a fantastic way to verify calculations and understand concepts more deeply.

Sine Function

The sine function, denoted as \( y = \sin x \), is one of the most well-known trigonometric functions. It describes a smooth, continuous wave that repeats every \( 2\pi \), known as the period of the sine function. Within each period:

• The function begins at zero.
• It reaches a maximum value of 1 at \( \frac{\pi}{2} \), the quarter cycle.
• It returns to zero at \( \pi \), completing a half cycle.

The behavior of the sine function is defined as:

• Within \( [0, \pi] \): It rises from 0 to 1 and falls back to 0, forming a single arch.
• Beyond \( \pi \): The cycle repeats in the negative direction, then back upwards.

The sinuous oscillation of the sine curve finds applications not only in mathematics but also in physics, engineering, and even music theory where waveform representation is crucial.

Understanding the properties of the sine function can immensely help in evaluating areas under curves and solving real-world problems where periodicity is involved.

Riemann Integrals

Riemann integrals provide a method to calculate the exact area under curves, which is fundamental to integral calculus. The concept rests on approximating the area by dividing it into small, easily calculable shapes, typically rectangles, and then summing these areas.

• Consider tiny intervals over the range \( [0, \pi] \) for the function \( \sin x \).
• Evaluate the height of the sine curve at multiple small points within each interval.
• The sum of these rectangle areas closely estimates the actual area under the curve, improving as the rectangles get narrower.

To find the exact area beneath \( y = \sin x \) from \( 0 \) to \( \pi \), you would use the definite integral:\[\int_0^{\pi} \sin x \, dx\]The solution involves integrating \( \sin x \), which results in proper calculations and yields an exact area of 2.

Riemann sums, which inform Riemann integrals, bridge the gap between discrete estimation and continuous calculation, enlightening how accumulation can be precisely described using calculus.