Problem 44
Question
Find the area under the curve for each function and interval given, using the rectangle method and \(n\) subintervals of equal width. $$f(x)=-\frac{1}{2} x^{3}+6 x, x \in[0,3]$$
Step-by-Step Solution
Verified Answer
Approximate the area using Riemann sums by computing the sum of rectangle areas for a chosen \( n \).
1Step 1: Understand the Rectangle Method
The rectangle method, also known as the Riemann sum method, is used to approximate the area under a curve by dividing the area into rectangles. For this problem, we will use the right endpoint method.
2Step 2: Identify the Function and Interval
The function given is \( f(x) = -\frac{1}{2}x^3 + 6x \) and the interval is \([0, 3]\). We need to compute the area under this curve using \( n \) rectangles.
3Step 3: Calculate the Width of Each Subinterval
The width of each rectangle, denoted as \( \Delta x \), is calculated by dividing the length of the interval by \( n \). Thus, \( \Delta x = \frac{3 - 0}{n} = \frac{3}{n} \).
4Step 4: Determine the Right Endpoints
For each rectangle, the right endpoint will be at \( x_i = i\cdot \Delta x \) where \( i \) ranges from 1 to \( n \). The right endpoint of each subinterval is needed to calculate the height of the rectangles.
5Step 5: Compute the Height of Each Rectangle
The height of each rectangle is given by the value of the function at the right endpoint of the subinterval: \( f(x_i) = -\frac{1}{2}(x_i)^3 + 6(x_i) \).
6Step 6: Calculate the Sum of the Areas of the Rectangles
Sum up the areas of all rectangles which can be computed as \( A \approx \sum_{i=1}^{n} f(x_i) \Delta x \). Substitute \( f(x_i) \) and \( \Delta x = \frac{3}{n} \) into this sum.
7Step 7: Derive the Expression for the Approximate Area
The approximate area \( A \) is \( \sum_{i=1}^{n} \left(-\frac{1}{2}\left(\frac{3i}{n}\right)^3 + 6\left(\frac{3i}{n}\right)\right) \frac{3}{n} \). Simplify this expression to have a clearer function for different values of \( n \).
8Step 8: Calculate Specific Value for \( n \)
Select a specific \( n \), say \( n = 6 \), and compute the associated area. Plug \( n = 6 \) into the area expression to get the approximate numerical value.
Key Concepts
Area Under a CurveRectangle MethodRight Endpoint Approximation
Area Under a Curve
Finding the area under a curve is a way of quantifying the space beneath a curve on the graph of a function across a given interval. This is particularly useful in mathematics and sciences to represent accumulated quantities, such as total distance or probability.
In this particular problem, the goal is to determine the accumulated area from the function described by the polynomial \( f(x) = -\frac{1}{2}x^3 + 6x \) over the interval \([0, 3]\). The area measures how much "space," in units squared, lies between the curve and the x-axis. It's as if you are computing the amount of paint needed to cover the ground under a wavy roof.
Since curve areas often don't have neat geometric shapes, we typically use methods that allow us to approximate the area unless a definite formula can be derived. The Riemann sum method, involving rectangles like in this exercise, is such an approach commonly employed for approximation. It helps break down complex areas into simple, manageable shapes.
In this particular problem, the goal is to determine the accumulated area from the function described by the polynomial \( f(x) = -\frac{1}{2}x^3 + 6x \) over the interval \([0, 3]\). The area measures how much "space," in units squared, lies between the curve and the x-axis. It's as if you are computing the amount of paint needed to cover the ground under a wavy roof.
Since curve areas often don't have neat geometric shapes, we typically use methods that allow us to approximate the area unless a definite formula can be derived. The Riemann sum method, involving rectangles like in this exercise, is such an approach commonly employed for approximation. It helps break down complex areas into simple, manageable shapes.
Rectangle Method
The rectangle method, also referred to as the Riemann sum method, is a technique used to estimate the area under a curve. This technique involves dissecting the area into a certain number of small rectangles. Each rectangle captures a portion of the area, and by summing their areas, we approximate the total area.
In the context of this exercise, the curve defined by \( f(x) = -\frac{1}{2}x^3 + 6x \) on \([0, 3]\) is divided into \(n\) subintervals of equal width. The width of each rectangle, \( \Delta x \), is calculated by dividing the entire length of the interval by \(n\), such that \( \Delta x = \frac{3}{n} \).
By selecting the right endpoints for each rectangle's height, the method captures the function's behavior over each subinterval to better grasp changes and variations in its shape, thus leading to a comprehensive approximation of the intended area.
In the context of this exercise, the curve defined by \( f(x) = -\frac{1}{2}x^3 + 6x \) on \([0, 3]\) is divided into \(n\) subintervals of equal width. The width of each rectangle, \( \Delta x \), is calculated by dividing the entire length of the interval by \(n\), such that \( \Delta x = \frac{3}{n} \).
By selecting the right endpoints for each rectangle's height, the method captures the function's behavior over each subinterval to better grasp changes and variations in its shape, thus leading to a comprehensive approximation of the intended area.
Right Endpoint Approximation
The right endpoint approximation is a specific type of Riemann sum used when applying the rectangle method for area calculations. It determines the height of each rectangle by evaluating the function at the right-hand side of each subinterval. This is crucial since the height of each rectangle essentially decides how much area we account for when approximating.
During the approximation process for this exercise, each rectangle's height is calculated with the function \( f(x_i) = -\frac{1}{2}(x_i)^3 + 6(x_i) \). Here, \( x_i \) denotes the right endpoint of the subinterval, defined as \( x_i = i \cdot \Delta x \), with \( i \) running from 1 through to \( n \).
As you increase \( n \), making rectangles narrower and potentially an infinite number, this approximation improves. Trying different values of \( n \) gives insight into how closely it can match the actual area. The right endpoint method ensures all curve peaks and valleys are captured to best recognize the actual area beneath the curve.
During the approximation process for this exercise, each rectangle's height is calculated with the function \( f(x_i) = -\frac{1}{2}(x_i)^3 + 6(x_i) \). Here, \( x_i \) denotes the right endpoint of the subinterval, defined as \( x_i = i \cdot \Delta x \), with \( i \) running from 1 through to \( n \).
As you increase \( n \), making rectangles narrower and potentially an infinite number, this approximation improves. Trying different values of \( n \) gives insight into how closely it can match the actual area. The right endpoint method ensures all curve peaks and valleys are captured to best recognize the actual area beneath the curve.
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Problem 44
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