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Chapter 11: Problem 57

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

Short Answer

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

Step by step solution

01

Set Up The Integral

Write the given integral \( \int_{0}^{2}(2-x)\sqrt{x} \, dx \) in a standard form, suitable for calculations.
02

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 \(\frac{4}{3} x^{\frac{3}{2}}\Big|_{0}^{2}\).
03

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 \(\frac{16}{15}\).
04

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.

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Most popular questions from this chapter

Use the Trapezoidal Rule with \(n=4\) to approximate the definite integral. $$ \int_{0}^{2} \frac{1}{x+1} d x $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(x)=\frac{1}{x}, g(x)=-e^{x}, x=\frac{1}{2}, x=1 $$

A company purchases a new machine for which the rate of depreciation can be modeled by \(\frac{d V}{d t}=10,000(t-6), \quad 0 \leq t \leq 5\) where \(V\) is the value of the machine after \(t\) years. Set up and evaluate the definite integral that yields the total loss of value of the machine over the first 3 years.

Use the Trapezoidal Rule with \(n=4\) to approximate the definite integral. $$ \int_{0}^{4} \sqrt{1+x^{2}} d x $$

Use a graphing utility to graph the region bounded by the graphs of the functions. Write the definite integrals that represent the area of the region. (Hint: Multiple integrals may be necessary.) $$ y=\frac{4}{x}, y=x, x=1, x=4 $$
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