Question

Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=2 x^{2} $$

Short answer

The Midpoint Rule is used to approximate the area under the curve of a function and can be calculated using the midpoint of each partition. The exact area under the curve can be calculated using integration over the interval. The two can be compared to assess the accuracy of the Midpoint Rule. The region can be sketched by drawing the function and shading the desired area.

Step-by-step solution

Understanding Midpoint Rule

The Midpoint Rule is a method for approximating the definite integral of a function. It does this by dividing the area under the curve into rectangles and summing the areas of these rectangles. We're given that the number of rectangles (or partitions), \(n\), is 4.

Determine Width of Partitions

The width of each partition, \(Δx\), can be calculated by subtracting the lower bound from the upper bound and dividing by \(n\). Assuming our interval is from \(0\) to \(1\), the width \(Δx = (1 - 0)/4 = 0.25\).

Midpoint Rule Approximation

To use the Midpoint Rule, we calculate the value of the function at the midpoint of each interval and then multiply it by the width of the interval. The midpoints for our intervals are \(0.125, 0.375, 0.625,\) and \(0.875\). We plug these into our equation, \(f(x) = 2x^2\), to find the heights of the rectangles and then multiply each by \(Δx=0.25\).

Compute Midpoint Rule Approximation

Now we add up the areas of the rectangles, \(f(0.125)*0.25 + f(0.375)*0.25 + f(0.625)*0.25 + f(0.875)*0.25\), and calculate to get our approximation.

Exact Area

To get the exact area under the curve, integrate the function \(f(x) = 2x^2\) from \(0\) to \(1\). This will be our benchmark to compare the Midpoint Rule approximation.

Compare Approximations

Compare the value obtained in step 4 (Midpoint approximation) with the exact area obtained in step 5. This gives us the degree of accuracy of the Midpoint Rule approximation.

Sketch the Region

Draw the graph of \(f(x) = 2x^2\) from \(0\) to \(1\) and shade the area under the curve within this interval. This region represents the exact area. The rectangles that have been calculated represent the Midpoint Rule approximation of the area.

Key concepts

Definite Integral

Definite integrals are used to calculate the area under a curve between two points on a function, typically on the x-axis. When we integrate a function over an interval, we are summing an infinite number of infinitesimally small products of function values and widths, which give us the total area under the curve within those bounds.
For instance, with the function \( f(x) = 2x^2 \), integrating it from \(0\) to \(1\) provides the exact area under the curve between these points. This process essentially accumulates all the small slices or strips of area to get the total sum.
The Midpoint Rule leverages the concept of definite integration to approximate this area by breaking it into a finite number of segments, rather than taking an infinite sum.

Function Approximation

In many cases, calculating a definite integral directly can be complex or impossible analytically. This is where function approximation methods like the Midpoint Rule come in handy. They allow us to estimate the area under a curve using simpler calculations.
The idea behind these methods is to approximate the curve with simpler geometric shapes whose areas are easy to calculate. In the case of the Midpoint Rule, rectangles are used to approximate the area.

• This method is particularly useful in applied contexts where exact values aren't necessary or function behavior isn't fully known.
• It provides a practical means to get a close estimate of integrals for experimental data or non-standard functions.

In summary, function approximation is all about making complex mathematical problems manageable and solvable with basic arithmetic.

Rectangles

Rectangles play a crucial role in the Midpoint Rule, serving as the primary shape for estimating areas under curves. By dividing an interval into equal parts, rectangles are formed, each representing a segment of the area under the curve.
Each rectangle's width is determined by the division of the total interval length by the number of rectangles desired. The height of each rectangle is determined by evaluating the function at the midpoint of the interval it covers.

• The area of each rectangle is then calculated as the product of its width and height.
• These areas are summed to approximate the total area under the curve.

The accuracy of this approximation depends on the number of rectangles used; more rectangles usually mean a better approximation.

Area Under Curve

The area under a curve, in mathematical terms, represents the integral of a function over a given interval. This measurement is critical in fields like physics and engineering, where it might represent the total quantity or cumulative value of a rate measure over time or space.
Specifically, for the function \( f(x) = 2x^2 \) from \(0\) to \(1\), the area under the curve can be thought of as the exact quantity that the Midpoint Rule aims to approximate.
By visualizing this area with methods like sketching or computational software, one can get a more intuitive understanding of how the function accumulates over an interval.

• Calculating the area through the Midpoint Rule involves seeing the curve as a series of flat, manageable segments (rectangles).
• This offers a straightforward visual and numerical way to grasp how changes in the function reflect on the total area.

Understanding the area under a curve is essential for interpreting how functions relate to real-world phenomena.