Problem 54 - Essential Calculus

Question

In Exercises \(53-56,\) use the Midpoint Rule Area \(\approx \sum_{i=1}^{n} f\left(\frac{x_{i}+x_{i-1}}{2}\right) \Delta x\) with \(n=4\) to approximate the area of the region bounded by the graph of the function and the \(x\) -axis over the given interval. $$ f(x)=x^{2}+4 x, \quad[0,4] $$

Step-by-Step Solution

Answer

The approximate area under the curve is 53 units.

Step 1: Interpret the interval [0,4]

Firstly, the given interval is [0,4]. This interval can be divided into 4 equal parts because \(n=4\). When these 4 parts are calculated, it results to \(\Delta x = (4 - 0) / 4 = 1 \) . This means that the subintervals are [0,1], [1,2], [2,3] and [3,4].

Step 2: Applying the Midpoint Rule

On each subinterval, the Midpoint Rule tells to calculate the function at the middle of each interval, multipy it with \(\Delta x\), and finally to sum up all those values. Therefore, calculate for the midpoints which are at \(0.5, 1.5, 2.5\) and \(3.5\). Substituting these midpoints into the function: \(f(0.5)=0.5^2+4 (0.5)=2.25\), \(f(1.5)=1.5^2+4(1.5)=8.25\), \(f(2.5)=2.5^2+4 (2.5)=16.25\), \(f(3.5)=3.5^2 +4 (3.5)=26.25\).

Step 3: Summing the results

According to the Midpoint Rule, the total sum would be \(\Delta x\) multiplied by the sum of the function values at midpoints. Therefore, the area would be \(Area \approx \Delta x [f(0.5)+f(1.5)+f(2.5)+f(3.5)]= 1(2.25+8.25+16.25+26.25)=53\).

Key Concepts

Numerical Integration

Numerical integration is a technique to approximate the definite integral of a function, especially when finding its exact form is challenging or impossible. To evaluate the integral, we break it into simpler, calculable parts. This method proves useful when dealing with complicated functions or intervals.
In practical terms, numerical integration is about finding the area under a curve. Methods like the Trapezoidal Rule and Simpson's Rule also fall under this category. However, in our case, the Midpoint Rule is used, which relies on taking the midpoint of each subinterval to estimate the function's behavior.

• Subintervals: The entire interval of interest is divided into smaller parts.
• Midpoints: We calculate the value of the function at the midpoint of each subinterval.
• Sum: Aggregate the contributions of each subinterval to estimate the overall area.

Definite Integral

The definite integral gives the accumulated value, such as area or volume, over a specific interval. It is denoted by the symbol \int_{a}^{b} f(x) \, dx\, where \ a \ and \ b \ are the limits of integration.
Over the interval \[0,4\]\, the definite integral corresponds to computing the total area under the curve of \ f(x) = x^2 + 4x \ and above the x-axis.
Calculating this exactly can be complex, hence the need for approximation techniques like numerical integration.
Understanding integrals as accumulated sums over an interval helps in visualizing how function values contribute to quantities like areas beneath curves.

• Limits of Integration: Define the start and end points of the interval.
• Accumulation: Represents the total contribution of all subintervals.

Approximation Methods

Due to the complexity of some functions, it is often necessary to use approximation methods to find their integrals. These methods give us a close estimate of what the exact calculation would yield.
The Midpoint Rule is one such method, chosen for its simplicity and ability to offer reasonable accuracy by using midpoints of subintervals for calculation.
For functions that change rapidly within an interval, approximation approaches such as the Midpoint Rule allow us to gain insights about the integral's value without needing exact solutions.
Approximation methods work especially well in computers, where the process of dividing intervals and adding up results can be automated efficiently.

• Efficiency: Offers a quick approach to evaluate complex integrals.
• Robustness: Tolerates variations in interval selection and computation.
• Simplicity: Easy to understand and implement, especially for case studies like this.