Question

Use the Midpoint Rule with \(n=4\) to approximate the area of the region. Compare your result with the exact area obtained with a definite integral. $$ f(y)=y^{2}+1, \quad[0,4] $$

Short answer

First, compute the Midpoint Rule approximation by dividing the interval into four equal subintervals, and then evaluate the function at the midpoint of each subinterval. Secondly, calculate the exact area using the definite integral of the function. Finally, compare both results to see how close the approximation is to the actual result.

Step-by-step solution

Define the function

The given function is \( f(y) = y^{2}+1 \). The curve of this function is the area that needs to be found from \(y = 0\) to \(y = 4\).

Apply the Midpoint Rule

The Midpoint Rule formula is \( ∑_{i=1}^{n}f((x_{i-1}+x_i)/2) * Δx \) . In this exercise, \(n = 4\), therefore four rectangles will be used to approximate the area under the curve. The width of each rectangle, \(Δy = (b - a)/n = (4 - 0)/4 = 1\). Now, find the \(y\) coordinates of the midpoints of each subinterval or rectangle, which are \(0.5, 1.5, 2.5, 3.5\) and substitute them to the function. After these calculations, add up all the resulting function evaluations and multiply by the width of the rectangles.

Calculate the exact area

The exact area under the curve is given by the definite integral of the function on the interval \([0, 4]\). This is found by evaluating the integral \(∫_{0}^{4} (y^{2}+1) dy\).

Comparison of results

Compare the approximation obtained with the Midpoint Rule and the exact result from the integral solution. This comparison will give an understanding of how precise the Midpoint Rule was in this case.