Problem 57 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the definite integral by hand. Then use a graphing utility to graph the region whose area is represented by the integral. $$ \int_{0}^{2}(2-x) \sqrt{x} d x $$

Step-by-Step Solution

Answer

The value of the definite integral \( \int_{0}^{2}(2-x) \sqrt{x} \, dx \) is \(\frac{16}{15}\).

Step 1: Set Up The Integral

Write the given integral \( \int_{0}^{2}(2-x)\sqrt{x} \, dx \) in a standard form, suitable for calculations.

Step 2: Solve the Integral

The integral can be divided into two integrals and further simplified as \( - \int_{0}^{2} x \sqrt{x} \, dx + \int_{0}^{2} 2 \sqrt{x} \, dx \). Now, proceed with computing each integral separately. Using the power rule for integration, the first part becomes \(- \frac{2}{5} x^{\frac{5}{2}}\Big|_{0}^{2}\) and second part becomes \(\frac{4}{3} x^{\frac{3}{2}}\Big|_{0}^{2}\).

Step 3: Final Evaluation

Apply the limits of the integral by substituting the upper limit (2) and then subtracting the result of substituting the lower limit (0). That gives us a final answer of \(- \frac{16}{5} + \frac{16}{3}\) which simplifies to \(\frac{16}{15}\).

Step 4: Graphing the Area

The graph of the function \( (2-x)\sqrt{x} \) from x=0 to x=2 represents a region in the first quadrant. The area under the curve of this function within these limits is represented by the value of the definite integral. Use a graphing utility to graph this function, and shade the area under the curve between x=0 and x=2 to represent the result of the integral. Please note that graphing is best done visually with a utility and might be outside the scope of this text-based explanation.