Problem 19 - A Course in Calculus and Real Analysis

Question

Let \(f:[0, \infty) \rightarrow \mathbb{R}\) be continuous and \(f(x) \geq 0\) for all \(x \in[0, \infty)\). If for each \(b>0\), the area bounded by the \(x\) -axis, the lines \(x=0, x=b\), and the curve \(y=f(x)\) is given by \(\sqrt{b^{2}+1}-1\), determine the function \(f\).

Step-by-Step Solution

Answer

The function \(f\) is given by \(f(x) = \frac{x}{\sqrt{x^2 + 1}}\).

Step 1: Express the area under the curve in terms of the function f

We are given that the area under the curve \(y = f(x)\), starting from \(x = 0\) to \(x = b\) is equal to \(\sqrt{b^2 + 1} - 1\). We can express this area using the definite integral of the function \(f(x)\) from \(x = 0\) to \(x = b\), which can be written as follows: \[\int_{0}^{b} f(x) dx = \sqrt{b^2 + 1} - 1\]

Step 2: Differentiate both sides of the equation with respect to b

In order to find the function \(f\), we can differentiate both sides of the above equation with respect to \(b\). Using the Fundamental Theorem of Calculus, the derivative of the left-hand side of the equation is simply \(f(b)\). We will use the chain rule to differentiate the right-hand side of the equation: \[\frac{d}{db} \int_{0}^{b} f(x) dx = \frac{d}{db}(\sqrt{b^2 + 1} - 1)\] \[f(b) = \frac{b}{\sqrt{b^2 + 1}}\]

Step 3: Express f(x) in terms of x

Now that we have an expression for \(f(b)\), we can express \(f(x)\) in terms of \(x\) by simply replacing \(b\) with \(x\): \[f(x) = \frac{x}{\sqrt{x^2 + 1}}\] So, the function \(f\) is given by: \[f(x) = \frac{x}{\sqrt{x^2 + 1}}\]