Problem 43 - Precalculus

Question

Find the area under the curve for each function and interval given, using the rectangle method and \(n\) subintervals of equal width. $$f(x)=-\frac{1}{2} x^{2}+4 x, x \in[0,6]$$

Step-by-Step Solution

Answer

Use the midpoint rectangle method with \(n\) subintervals for the function within the interval \([0,6]\) to approximate the area.

Step 1: Determine Subinterval Width

The interval of integration is \([0, 6]\), meaning the total length of the interval is 6. Given \(n\) subintervals, each subinterval will have a width of \(\Delta x = \frac{6}{n}\).

Step 2: Identify Midpoint of Each Subinterval

For the midpoint rectangle method, the midpoint \(x_i\) of each subinterval \([x_{i-1}, x_i]\) is given by \(x_i = x_{i-1} + \frac{\Delta x}{2}\), where \(x_{i-1} = (i-1)\Delta x\). Calculate \(x_i\) for each subinterval using these formulas.

Step 3: Evaluate the Function at Each Midpoint

Calculate the value of the function \(f(x) = -\frac{1}{2}x^2 + 4x\) at each midpoint \(x_i\). This gives the height of each rectangle: \(f(x_i)\).

Step 4: Calculate Area of Each Rectangle

The area of each rectangle is given by the product of the height and the width: \(\text{Area}_i = f(x_i) \cdot \Delta x\). Sum these areas over all subintervals.

Step 5: Sum the Areas of All Rectangles

The total approximate area under the curve is the sum of the areas of all \(n\) rectangles, \(\sum_{i=1}^{n} \text{Area}_i = \sum_{i=1}^{n} f(x_i) \cdot \Delta x\).

Step 6: Calculate the Approximate Area

Substitute step 2 and 3 expressions in the summation from step 5. Simplify and calculate to find the total approximate area under the curve.

Key Concepts

Subinterval Width

The concept of subinterval width is crucial when using the rectangle method for approximating areas under a curve. To divide an interval into equal parts with clear boundaries, we determine the width, Δx, of each subinterval. Consider an interval [0,6], split into nequally sized subintervals. The width of each subinterval is given by the formula: Δx = \( \frac{6}{n} \). This ensures that the entire range from 0 to 6 is covered evenly.

• Each subinterval has the same width of Δx.
• Knowing the width helps in further calculations, including midpoint determination and area approximation.

Having a consistent subinterval width is foundational to accurately estimating the function's behavior across the interval.

Midpoint Approximation

Midpoint approximation is a key step in utilizing the rectangle method for area estimation. Each rectangle’s height is determined by evaluating the function at the midpoint of each subinterval. Here's how to find the midpoints: the midpoint x_iof each subinterval [x_{i-1}, x_i] is calculated using:x_i = x_{i-1} + \( \frac{Δx}{2} \), where x_{i-1} = (i-1)Δx.
This method ensures that the function's value at the center of each subinterval is used, which typically provides a better approximation than using either endpoint.

• The midpoint offers a balance, taking into account the function’s behavior across the subinterval.
• Accurate midpoints lead to more reliable area estimations.

Midpoint approximation thus plays a crucial role by determining points where the function is evaluated to approximate area.

Function Evaluation at Midpoints

Evaluating the function at midpoints involves substituting these points into the given function to determine each rectangle's height under the curve. For the function f(x) = -\( \frac{1}{2}x^2 + 4x \), we compute f(x_i)at each midpoint x_i.This gives us the height of the rectangle corresponding to each subinterval.

• The calculation of f(x_i)provides the rectangle’s height.
• A correct evaluation ensures the approximation closely represents the curve’s shape.

It is this step that translates the abstract midpoint values into tangible measurements of height, vital for calculating each rectangle’s area.

Area Approximation

The final step in the rectangle method is the calculation of the approximate area under the curve.In this step, we multiply the height of each rectangle f(x_i)obtained from function evaluation at midpoints by the subinterval width Δx.This product yields the area of each individual rectangle:Area_i = \( f(x_i) \cdot Δx \).
To find the total approximate area, sum all individual areas: \(\sum_{i=1}^{n} f(x_i) \cdot Δx\).

• This summation provides a practical estimate of the integral over the interval [0,6].
• A greater number of subintervals (n)usually results in a more accurate approximation.

Through these calculations, the rectangle method offers an effective technique for estimating areas where exact calculus solutions might be complex or unavailable.