"Use symmetry to evaluate the following integrals. ∫−22(x2+x3)dx" was answered by our experts

High School
Calculus and Analysis
Calculus 2
Applications of integrals

diferira7c

Answered question

2022-01-03

Use symmetry to evaluate the following integrals.

\int_{- 2}^{2} (x^{2} + x^{3}) d x

Answer & Explanation

Mason Hall

Beginner2022-01-04Added 36 answers

Step 1
Given: An integral

\int_{- 2}^{2} (x^{2} + x^{3}) d x

By symmetry an integral of form

\int_{- a}^{a} f (x) d x

can be written as

\int_{- a}^{0} f (x) d x + \int_{0}^{a} f (x) d x

Therefore,

\int_{- 2}^{2} (x^{2} + x^{3}) d x

can be written as:

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

Step 2
Now integrating the above function with respect to x.

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

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

= [(0 + 0) - (\frac{- 8}{3} + 4)] + [(\frac{8}{3} + 4) - (0 + 0)]

= \frac{8}{3} - 4 + \frac{8}{3} + 4

= \frac{16}{3}

Thus, the value of

\int_{- 2}^{2} (x^{2} + x^{3}) d x

\frac{16}{3}

Lynne Trussell

Beginner2022-01-05Added 32 answers

\int (x^{2} + x^{3}) d x

Lets

karton

Expert2022-01-11Added 613 answers

Given:
$\int_{- 2}^{2} (x^{2} + x^{3}) d x \int x^{2} + x^{3} d x \int x^{2} d x + \int x^{3} d x \frac{x^{3}}{3} + \frac{x^{4}}{4} (\frac{x^{3}}{3} + \frac{x^{4}}{4}) |_{- 2}^{2} \frac{2^{3}}{3} + \frac{2^{4}}{4} - (\frac{(- 2)^{3}}{3} + \frac{(- 2)^{4}}{4}) A n s w e r : \frac{16}{3}$

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