Question

Multiple Choice An LRAM sum with 4 equal subdivisions is used to approximate the area under the sine curve from \(x=0\) to \(x=\pi .\) What is the approximation? (A) \(\frac{\pi}{4}\left(0+\frac{\pi}{4}+\frac{\pi}{2}+\frac{3 \pi}{4}\right)\) B) \(\frac{\pi}{4}\left(0+\frac{1}{2}+\frac{\sqrt{3}}{2}+1\right)\) (C) \(\frac{\pi}{4}\left(0+\frac{\sqrt{2}}{2}+1+\frac{\sqrt{2}}{2}\right)\) D) \(\frac{\pi}{4}\left(0+\frac{1}{2}+\frac{\sqrt{2}}{2}+\frac{\sqrt{3}}{2}\right)\) (E) \(\frac{\pi}{4}\left(\frac{1}{2}+\frac{\sqrt{2}}{2}+\frac{\sqrt{3}}{2}+1\right)\)

Short answer

The approximation for the area under the curve using the LRAM sum with 4 equal subdivisions is \( \frac{\pi}{4}\left(0+\frac{1}{\sqrt{2}}+1+\frac{1}{\sqrt{2}}\right)\). Thus, option (C) is the correct answer.

Step-by-step solution

Determine the left endpoints of the subdivisions

The interval from 0 to \( \pi \) is divided into 4 equal parts, meaning the endpoints of these subdivisions are 0, \(\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}\), and \( \pi \). As we're using the Left Rectangle Approximation Method (LRAM), we consider the left endpoints for the area approximation, which are: 0, \(\frac{\pi}{4}, \frac{\pi}{2}, \frac{3\pi}{4}\).

Find the sine value of each endpoint

Since we're working with a sine curve, the height of each left rectangle will be the sine of the left endpoint. Calculate the sine values: \(\sin(0) = 0, \sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}, \sin(\frac{\pi}{2}) = 1, \sin(\frac{3\pi}{4}) = \frac{\sqrt{2}}{2}\)

Calculate the approximation

The area under the curve is approximated by the sum of the areas of these rectangles. Each area is calculated by multiplying the width (\( \frac{\pi}{4} \)) of the rectangle by its height (the sine values calculated). This gives us the approximation \( \frac{\pi}{4}\left(0+\frac{\sqrt{2}}{2}+1+\frac{\sqrt{2}}{2}\right)\)

Key concepts

Sine Curve Integration

The process of integrating a sine function, known as sine curve integration, is a fundamental concept in calculus, especially when dealing with trigonometric functions. It involves finding the area under the sine curve between two points on the x-axis.

Since the sine function oscillates between -1 and 1, integrating it over a complete period (from 0 to \(2\pi\)) results in an area of zero because the positive and negative areas cancel out. However, when integrating from 0 to \(\pi\), you capture only the positive half of the sine wave, leading to a non-zero area.

To analytically integrate the sine function, we typically rely on the integral formula \[ \int \sin(x) dx = -\cos(x) + C \], where C is the integration constant. This formula allows the calculation of the exact area under the sine curve. However, in many practical situations, and especially in educational settings, we approximate this area using numerical methods such as the Left Rectangular Approximation Method (LRAM).

Rectangular Approximation Method

The Rectangular Approximation Method, or RAM, is a technique in the numerical analysis part of calculus used for estimating the area under a curve. One specific version of this method is the Left Rectangular Approximation Method (LRAM), where the area under a curve on a given interval is approximated using rectangles.

The method involves dividing the interval into a number of equal parts. For each subdivision, a rectangle is formed by taking the left point of the segment as the rectangle's height, which corresponds to the value of the function at that point. The width of each rectangle is the same and is determined by the length of the subdivisions.

In the case of LRAM, the sum of the areas of these rectangles provides an approximation of the area under the curve. While this method may not yield an exact result, it is particularly useful when dealing with functions that are difficult to integrate analytically or when a quick approximation is sufficient.

Calculus in Trigonometric Functions

Calculus in trigonometric functions often involves operations such as differentiation and integration applied to the sine, cosine, and other trigonometric functions. Understanding how to work with these functions is critical, as they appear frequently in both theoretical and applied mathematics.

For example, differentiation rules for trigonometric functions are as follows: the derivative of \(\sin(x)\) is \(\cos(x)\), and the derivative of \(\cos(x)\) is \( -\sin(x)\). When integrating, as mentioned before, \(\int \sin(x) dx = -\cos(x) + C\), and \(\int \cos(x) dx = \sin(x) + C\).

Being able to navigate these rules smoothly allows for solving more complex calculus problems that include trigonometric functions, such as finding the rates of change in harmonic motion or calculating work done by a varying force.

Area Under a Curve

The area under a curve in a graph represents the integral of the function within the specified bounds of the curve. This is a key concept in calculus that has practical applications in various fields, such as physics, economics, and engineering.

Finding the exact area requires integration, but when the integrals are challenging or impossible to solve analytically, approximation methods such as the LRAM come into play. These methods partition the area into simpler geometric shapes (like rectangles) and sum their areas for an approximation.

The accuracy of the approximation depends on the number and size of the rectangles; more rectangles typically lead to a better approximation. Even though it’s an approximation, understanding how to estimate the area under the curve with methods like the LRAM is an essential skill in calculus, offering a manageable approach to otherwise complex problems.