Problem 137 - Thomas Calculus in SI Units

Question

Find the area of the region between the curve \(y=2 x /\left(1+x^{2}\right)\) and the interval \(-2 \leq x \leq 2\) of the \(x\) -axis.

Step-by-Step Solution

Answer

The area is 0.

Step 1: Set Up the Integral

The area between the curve and the x-axis from \(x = -2\) to \(x = 2\) can be found using the integral: \[A = \int_{-2}^{2} \frac{2x}{1+x^2} \, dx\]

Step 2: Use Substitution Method

Perform a substitution to simplify the integral. Let \( u = 1 + x^2 \), then the derivative is \( du = 2x \, dx \). The limits of integration also change: when \(x = -2\), \(u = 5\), and when \(x = 2\), \(u = 5\). The integral becomes:\[\int_{5}^{5} \frac{1}{u} \, du\]

Step 3: Evaluate the Integral

Notice that the limits of integration are the same \(5\) to \(5\). When the upper and lower limits of an integral are the same, the value of the integral is zero.

Key Concepts

Integral Calculus

Integral Calculus is all about finding the total accumulation of some quantity. You can think of it like adding up small pieces to find out the whole picture. In math, integrals help us determine areas, volumes, central points, and many other critical figures.
In this exercise, we are focusing on finding the area under a curve, which is one of the essential applications of integral calculus. The integral, \(\int_{a}^{b} f(x) \, dx\), calculates this area, where \(f(x)\) is your function and \(a\) to \(b\) are your limits of integration.

• The integral symbol \(\int\) indicates summation.
• \