Question

For the data: \(t\) \(3 \quad 6\) \(9 \quad 12\) \(y(t)=-1\) 2 3 , the value of \(\int_{3}^{12} y(t)\) dt when eomputed by Simpson's \(\frac{1}{3} \mathrm{rd}\) rule is (a) 15 (b) 10 (c) 0 (d) \(5 .\)

Short answer

The computed integral value is approximately 11, which doesn't match any given option accurately.

Step-by-step solution

Identify the Function and Intervals

The function given is a piecewise linear function. We divide the interval from 3 to 12 into equal parts where data points are available. In this case, since we have four points: 3, 6, 9, and 12, this gives us three intervals of equal length, each of width 3.

Confirm the Use of Simpson's Rule

Simpson's Rule requires an even number of intervals. We have three intervals, fitting the requirement since Simpson's Rule can be applied using these four data points.

Apply Simpson's 1/3rd Rule Formula

The Simpson's 1/3rd rule formula is:\[\int_{a}^{b} y(t) \, dt \approx \frac{h}{3}(y_0 + 4y_1 + 2y_2 + 4y_3 + y_4)\]Here, the width of each interval, \(h\), is 3 (since the intervals are 3-6, 6-9, 9-12). The function values at the nodes are recorded as follows: \(y_0 = -1\), \(y_1 = -1\), \(y_2 = 2\), and \(y_3 = 3\).

Calculate the Integration

Substitute the values into the Simpson's 1/3rd rule formula:\[\int_{3}^{12} y(t) \, dt \approx \frac{3}{3}((-1) + 4(-1) + 2(2) + 4(3))\]Calculate the expression inside:\(-1 + 4(-1) + 2(2) + 4(3) = -1 - 4 + 4 + 12 = 11\).Thus,\[\int_{3}^{12} y(t) \, dt \approx 11\]

Determine the Correct Option

After calculating the approximate value using Simpson's 1/3rd rule, we obtain 11. The available options were 15, 10, 0, and 5. Since none of the options match precisely with the calculated value of 11, further double-checking of arithmetic might be necessary, or options might only stand for crude rounded estimates.

Key concepts

Numerical Integration

Numerical integration is a mathematical technique used to estimate the definite integral of a function when an exact solution is difficult or impossible to obtain analytically. It is particularly useful when dealing with real-world data or complex functions where traditional calculus methods are not suitable.
Numerical integration employs various algorithms to approximate the area under a curve over a specific interval. The most common methods include Trapezoidal Rule, Simpson's Rule, and Rectangular Methods. Each of these methods divides the interval into smaller sub-intervals, making the calculation more manageable.

• Trapezoidal Rule: Approximates the area under a curve using trapezoids.
• Simpson's Rule: Uses parabolic arcs instead of straight lines.
• Rectangular Methods: Involves summing up rectangular slices under the curve.

By dividing the area into manageable parts, numerical integration simplifies the problem and provides an effective approximation of the integral.

Piecewise Linear Function

A piecewise linear function is a function composed of straight-line segments, each defined over a specific interval. This type of function is often used when data points are available, leading to a simple and effective representation of changing data. In our example, the function values change suddenly at defined points rather than gradually.
A piecewise linear function can be advantageous due to its simplicity. It's often used in real-world applications where data is collected at discrete intervals, such as economics, physics, or engineering.

• Advantages: Easy to compute and integrate over given intervals.
• Use Cases: Modeling time-dependent or segmented systems.
• Characteristics: Sudden changes at segment boundaries, making it straightforward to calculate integral values between specific points.

Understanding piecewise linear functions help clarify how numerical methods like Simpson's Rule can be applied efficiently.

Intervals in Integration

Intervals in integration are the specific ranges over which an integral is calculated. These intervals are crucial as they define the section of the function being analyzed. In numerical integration, such as with Simpson's Rule, the interval is typically divided into smaller equal parts to aid calculation.
For the exercise, the interval from 3 to 12 is divided into three equal parts. This gives us intervals of width 3, namely [3,6], [6,9], and [9,12]. Properly executed intervals are significant for accurate estimation as they dictate the number of calculations required and the precision of the approximation.

• Equal Intervals: Necessary for many numerical methods like Simpson's Rule.
• Determination: Defined by the data points in question, leading to a clearer understanding of function behavior within each segment.
• Impact on Accuracy: More intervals usually result in better precision.

In this exercise, dividing the interval correctly facilitated the application of Simpson's Rule.

Quadrature Formulas

Quadrature formulas are a set of techniques used to approximate the definite integral of a function. These formulas break down integration into a sum of weighted function values at specified points, offering a reliable way to evaluate integrals numerically. Simpson's Rule is a specific example of a quadrature formula.
Simpson's Rule, in particular, uses parabolic segments to approximate the area under a curve over an interval. It is especially effective for functions that can be approximated well by parabolas.

• Formula: Uses weights of 1, 4, and 2 for successive segments.
• Requirements: Requires an even number of intervals.
• Advantages: Generally more accurate than methods using linear approximations, like the Trapezoidal Rule.

Understanding these formulas helps students apply them with confidence, ensuring they can approximate integrals accurately. In this exercise, the use of Simpson’s Rule as a quadrature formula offered a more precise result than other simpler approaches.