Question: "Determine the following definite integrals: ∫−13|2x−x2|dx"

aflacatn

Answered question

2021-09-16

Determine the following definite integrals:

\int_{- 1}^{3} | 2 x - x^{2} | d x

avortarF

Skilled2021-09-17Added 113 answers

Step 1
Evaluate the integrals:

f (x) = \int_{- 1}^{3} | 2 x - x^{2} | d x

Eliminate the absolutes,

- 1 \leq x \leq 0 f (x) = (- 2 x + x^{2})

0 \leq x \leq 2 f (x) = (2 x - x^{2})

2 \leq x \leq 3 f (x) = (- 2 x + x^{2})

f (x) = \int_{- 1}^{0} (- 2 x + x^{2}) d x + \int_{0}^{2} (2 x - x^{2}) d x + \int_{2}^{3} (- 2 x + x^{2}) d x

= {[- 2 (\frac{x^{2}}{2}) + \frac{x^{3}}{3}]}_{- 1}^{0} + {[2 (\frac{x^{2}}{2}) - \frac{x^{3}}{3}]}_{0}^{2} + {[- 2 (\frac{x^{2}}{2}) + \frac{x^{3}}{3}]}_{2}^{3}

= {[- x^{2} + \frac{x^{3}}{3}]}_{- 1}^{0} + {[x^{2} - \frac{x^{3}}{3}]}_{0}^{2} + {[- x^{2} + \frac{x^{3}}{3}]}_{2}^{3}

= [0 - (- {(- 1)}^{2} + \frac{{(- 1)}^{3}}{3})] + [2^{2} - \frac{2^{3}}{3} - 0] + [(- 3^{2} + \frac{3^{3}}{3}) - (- 2^{2} + \frac{2^{3}}{3})]

This is helpful 16 Flag

Jeffrey Jordon

Expert2021-11-08Added 2605 answers

Answer is given below (on video)

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