Problem 14 - Essential Calculus

Question

Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{-8}^{-1} \frac{x-x^{2}}{2 \sqrt[3]{x}} d x $$

Step-by-Step Solution

Answer

The value of the definite integral \( \int_{-8}^{-1} \frac{x-x^{2}}{2 \sqrt[3]{x}} d x \) is equal to \( \left(\frac{3}{10} (-1)^{\frac{5}{3}} - \frac{3}{10} (-1)^{\frac{5}{3}}\right) - \left(\frac{3}{10} (-8)^{\frac{5}{3}} - \frac{3}{10} (-8)^{\frac{5}{3}}\right)\)

Step 1: Simplify the function

Before integrating, it's easier to simplify the function first. Each term in the numerator is divided by \(2 \sqrt[3]{x}\). The simplified function is: \( \frac{x}{2 \sqrt[3]{x}} - \frac{x^{2}}{2 \sqrt[3]{x}} = \frac{1}{2} x^{\frac{2}{3}} - \frac{1}{2} x^{\frac{2}{3}} \)

Step 2: Find the antiderivative

Next, find the antiderivative of the function. The antiderivative of \(x^{\frac{2}{3}}\) is \( \frac{3}{5} x^{\frac{5}{3}}\) and the antiderivative of \(x^{\frac{2}{3}}\) is \( \frac{3}{5} x^{\frac{5}{3}}\). Incorporating the existing constants of \( \frac{1}{2} \), the antiderivative is \( F(x) = \frac{3}{10} x^{\frac{5}{3}} - \frac{3}{10} x^{\frac{5}{3}}\)

Step 3: Evaluate the definite integral

Finally, substitute the limits of integration into the antiderivative function and subtract. This gives \( \int_{-8}^{-1} \frac{x-x^{2}}{2 \sqrt[3]{x}} d x = F(-1) - F(-8) = \left(\frac{3}{10} (-1)^{\frac{5}{3}} - \frac{3}{10} (-1)^{\frac{5}{3}}\right) - \left(\frac{3}{10} (-8)^{\frac{5}{3}} - \frac{3}{10} (-8)^{\frac{5}{3}}\right)\)