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Chapter 4: Problem 13

Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. $$ \int_{0}^{1} \frac{x-\sqrt{x}}{3} d x $$

Short Answer

Expert verified
The value of the definite integral of the function \(\frac{x-\sqrt{x}}{3}\) from 0 to 1 is -1/2.

Step by step solution

01

Break Down the Function

First, it's important to see the function \(\frac{x-\sqrt{x}}{3}\) as the sum of two different functions divided by 3: \(\frac{x}{3}-\frac{\sqrt{x}}{3}\). This will make the integration easier since each function can be integrated separately.
02

Apply the Power Rule to each Function

Applying the power rule of integration to \(x/3\) we get \(\frac{x^2}{6}\) and to \(\sqrt{x}/3=\frac{x^{1/2}}{3}\) we get \(\frac{2}{3}x^{3/2}\). Remember, the power rule of integration states that the integral of \(x^n\) with respect to \(x\) is \(1/(n+1)*x^{n+1}\). Then, the definite integral from 0 to 1 of the function is obtained by evaluating the sum of these two antiderivatives at 1 and then subtracting the result of evaluating them at 0.
03

Evaluate the Definite Integral

The definite integral from 0 to 1 is obtained by evaluating \( (\frac{x^2}{6}-\frac{2}{3}x^{3/2})\) at x=1, which gives \( \frac{1}{6}-\frac{2}{3}\) and at x=0, which gives 0 (since any number multiplied by 0 is 0). The result is then \((\frac{1}{6}-\frac{2}{3}) - 0 = -\frac{1}{2}\).
04

Verify the Result with a Graphing Utility

As a final check, one can use a graphing utility to plot the function \(\frac{x-\sqrt{x}}{3}\) and verify that the area under the curve from 0 to 1 is indeed equal to -1/2. It should be noted that negative area corresponds to the portion of the graph that is below the x-axis.

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Most popular questions from this chapter

Linear and Quadratic Approximations In Exercises 33 and 34 use a computer algebra system to find the linear approximation \(P_{1}(x)=f(a)+f^{\prime}(a)(x-a)\) and the quadratic approximation \(P_{2}(x)=f(a)+f^{\prime}(a)(x-a)+\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}\) of the function \(f\) at \(x=a\). Use a graphing utility to graph the function and its linear and quadratic approximations. \(f(x)=\cosh x, \quad a=0\)

Evaluate, if possible, the integral $$\int_{0}^{2} \llbracket x \rrbracket d x$$

Find any relative extrema of the function. Use a graphing utility to confirm your result. \(f(x)=x \sinh (x-1)-\cosh (x-1)\)

Find the derivative of the function. \(y=\operatorname{sech}^{-1}(\cos 2 x), \quad 0

Find the derivative of the function. \(y=\left(\operatorname{csch}^{-1} x\right)^{2}\)
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