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angepepagiortzb

Answered question

2021-11-16

Write the integral as a sum of integrals without absolute values and evaluate.

\int_{- 2}^{3} | x^{3} | d x

Coon2000

Beginner2021-11-17Added 15 answers

Step 1
The given integral is:

\int_{- 2}^{3} | x^{3} | d x

By definition we have

| x | = {\begin{cases} x & i f x \geq 0 \\ - x & i f x \leq 0 \end{cases}

The interval [−2,3] is divided into two intervals [−2,0] and [0,3].
Step 2

\int_{- 2}^{3} | x^{3} | d x = \int_{- 2}^{0} - x^{3} d x + \int_{0}^{3} x^{3} d x

= {[\frac{- x^{4}}{4}]}_{- 2}^{0} + {[\frac{x^{4}}{4}]}_{0}^{3}

= - (0 - \frac{{(- 2)}^{4}}{4}) + (\frac{3^{4}}{4} - 0)

=24.25
Therefore the value of the integral is 24.25.

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