Problem 7 - College Algebra and Calculus: An Applied Approach

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. $$ f(x)=x^{2}+3 \quad[-1,1] $$

Step-by-Step Solution

Answer

The Midpoint Rule provides an approximation of the area under the curve \(f(x) = x^2 + 3\) over the interval [-1,1], which can then be compared with the exact area computed by evaluating the definite integral. The method is a numerical approximation, and the exactness of the result will vary.

Step 1: Understanding the exercise

The first step is to understand what is given and what is asked. Let's break it down. The exercise asks to use the Midpoint Rule with \(n=4\) to approximate the area under the curve \(f(x) = x^2 + 3\) on the interval [-1,1]. Then, compare the result with the exact area.

Step 2: Apply the Midpoint Rule

The Midpoint Rule states that \(\int_{a}^{b} f(x)dx \approx \Delta x [f(a+\frac{\Delta x}{2})+f(a + \frac{3\Delta x}{2})+f(a + \frac{5\Delta x}{2})+...+f(a + (2n-1)\frac{\Delta x}{2})]\), where \(\Delta x=(b-a)/n\). Let's apply this rule with \(a=-1, b=1, n=4\). This gives: \(\Delta x= \frac{1-(-1)}{4} = 0.5\), and then substitute the values into the Midpoint Rule formula to approximate the area.

Step 3: Compute the exact area

In order to compare the approximation to the exact area, it is necessary to evaluate the definite integral of the function \(f(x)\) from -1 to 1. The antiderivative of \(f(x) = x^2 + 3\) is \(F(x) = \frac{1}{3}x^3 + 3x\). This is evaluated at the endpoints of the interval, and the difference gives the exact area under the curve.

Step 4: Compare results and sketch the region

Having obtained both the approximation and the exact area, it is possible to compare them. They may not be identical - the Midpoint Rule is an approximation method that often underestimates, but it offers a good estimate of the exact area. Finally, the region bounded by the graph of \(f\) and the x-axis over the interval [-1, 1] can be sketched. This involves drawing the function \(f(x)=x^{2}+3\) and shading the area under the curve from x=-1 to x=1.