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. $$ f(x)=x^{2}+4 x \quad[0,4] $$

Short answer

The estimated area using the Midpoint Rule is 41 square units while the exact area under the curve is 32 square units. Therefore, the Midpoint Rule estimate is reasonably close but overestimates the actual area.

Step-by-step solution

Divide the interval into equal parts

The Midpoint Rule is a numerical integration technique for approximating the area under the curve of a function. According to the problem, \(n=4\), which means the interval needs to be divided into 4 equal parts. The given interval is [0,4]. Therefore, the width of each part is \( \Delta x = (4-0) / 4 = 1 \) . The four sub intervals are [0,1], [1,2], [2,3], and [3,4].

Calculate the Midpoints

The midpoints of the intervals are where the function will be evaluated. For the intervals [0,1], [1,2], [2,3], and [3,4], the midpoints are 0.5, 1.5, 2.5, and 3.5, respectively.

Evaluate the function at the midpoints

The next step is to evaluate the function \( f(x)=x^{2}+4x \) at the midpoint of each interval. Therefore, calculate \( f(0.5), f(1.5), f(2.5), and f(3.5) \) . These numbers are found to be 2.25, 6.25, 12.25, and 20.25 respectively.

Calculate the Area of Each Rectangle

The area of each rectangle is the height times the width. The height is the value of the function at the midpoint and the width is \( \Delta x \) . Hence, the area of each rectangle are \( f(0.5) \Delta x = 2.25 , f(1.5) \Delta x = 6.25 , f(2.5) \Delta x = 12.25 , and f(3.5) \Delta x = 20.25 \).

Calculate the Approximate Total Area

Add up the areas of all the rectangles from Step 4 to get an approximation of the total area under the curve: \( 2.25 + 6.25 + 12.25 + 20.25 = 41 \) . This is an approximation of the total area using the Midpoint Rule.

Find the Exact Area

To find the exact area under the curve, calculate the definite integral of the function from 0 to 4: \( \int_{0}^{4}(x^{2}+4x)dx = [x^3/3 + 2x^2]_0^4 = 64/3 + 32 = 96/3 = 32 \) .

Compare the Midpoint Rule Approximation with the Exact Area

From the Midpoint Rule, approximation is 41, and the actual area under the curve is 32. It is noticed that the Midpoint Rule provides a good estimate, though not exact.