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Chapter 5: Problem 37

Evaluate the definite integral. \(\int_{0}^{4}\left(x^{1 / 2}+x^{1 / 4}\right) d x\)

Short Answer

Expert verified
\(\frac{2}{3}*4^{3/2} + \frac{4}{5}*4^{5/4} - \left(\frac{2}{3}*0^{3/2} + \frac{4}{5}*0^{5/4}\right) = 5.12\). So the answer of the integral is 5.12.

Step by step solution

01

Use the Power Rule for Integration

The power rule for integration states that \(\int x^n dx = \frac{1}{n+1} x^{n+1}\). Apply this rule for \(\int_{0}^{4}\left(x^{1 / 2}+x^{1 / 4}\right) d x\). It becomes \(\frac{2}{3}x^{3/2} + \frac{4}{5}x^{5/4}\) from 0 to 4.
02

Substitute the Upper Limit

Now substitute in the upper limit (4) into the equation, which yields \(\frac{2}{3}4^{3/2} + \frac{4}{5}4^{5/4}\)
03

Substitute the Lower Limit

Now substitute in the lower limit (0) into the equation, which yields \(\frac{2}{3}*0^{3/2} + \frac{4}{5}*0^{5/4}\)
04

Subtract the Results

Subtract the result from step 3 from the result of step 2 to obtain the final result.

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Most popular questions from this chapter

Evaluate the definite integral by hand. Then use a graphing utility to graph the region whose area is represented by the integral. \(\int_{2}^{4} \frac{3 x^{2}}{x^{3}-1} d x\)

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