Problem 12 - Essential Calculus

Question

Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{1}^{8} \sqrt{\frac{2}{x}} d x $$

Step-by-Step Solution

Answer

The value of the definite integral is \(8\sqrt{2} - 4\).

Step 1: Write down the integral in proper form

This step involves re-writing the integral in a proper form. Re-write the integral as: \(\int_{1}^{8} 2x^{-0.5} \, dx \). This makes the problem clearer to solve.

Step 2: Find the Antiderivative

Now we find the antiderivative F of the function \(2x^{-0.5}\). The antiderivative of a function can be found by reversing the rules of differentiation. Here, the antiderivative F of the given function is \(2 * 2x^{0.5}\) or \(4\sqrt{x}\).

Step 3: Use Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states that if \(F\) is an antiderivative of \(f\) on \([a, b]\), then \(\int_a^b f(x)\,dx=F(b)-F(a)\). We now apply this theorem using the antiderivative \(F = 4\sqrt{x}\), with \(a = 1\) and \(b = 8\). Thus, \(\int_{1}^{8} 2x^{-0.5}\,dx = F(8) - F(1) = 4\sqrt{8} - 4\sqrt{1}\).

Step 4: Simplify the result

Simplify the result of Step 3 to obtain the final value of the integral. That is, \(4\sqrt{8} - 4\sqrt{1} = 4 * 2\sqrt{2} - 4 = 8\sqrt{2} - 4\).