/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 78 Use the value \(\int_{0}^{2} x^{... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 78

Use the value \(\int_{0}^{2} x^{3} d x=4\) to evaluate each definite integral. Explain your reasoning. (a) \(\int_{-2}^{0} x^{3} d x\) (b) \(\int_{-2}^{2} x^{3} d x\) (c) \(\int_{0}^{2} 3 x^{3} d x\)

Short Answer

Expert verified
(a) \(\int_{-2}^{0} x^{3} dx = -4\), (b) \(\int_{-2}^{2} x^{3} dx = 0\), (c) \(\int_{0}^{2} 3x^{3} dx = 12\)

Step by step solution

01

Evaluate \(\int_{-2}^{0} x^{3} dx\)

The integral \(\int_{-2}^{0} x^{3} dx\) is the negative of \(\int_{0}^{2} x^{3} dx\) due to the symmetry of \(x^3\). Thus, if \(\int_{0}^{2} x^{3} dx = 4\), it follows that \(\int_{-2}^{0} x^{3} dx = -4\).
02

Evaluate \(\int_{-2}^{2} x^{3} dx\)

The integral \(\int_{-2}^{2} x^{3} dx\) can be broken down into two integrals as \(\int_{-2}^{2} x^{3} dx = \int_{-2}^{0} x^{3} dx + \int_{0}^{2} x^{3} dx\), applying the property of symmetry of the odd function we get this integral to be \(0\).
03

Evaluate \(\int_{0}^{2} 3x^{3} dx\)

The integral \(\int_{0}^{2} 3x^{3} dx\) is three times \(\int_{0}^{2} x^{3} dx\) due to the constant multiple rule of integrals. Since \(\int_{0}^{2} x^{3} dx = 4\), it follows that \(\int_{0}^{2} 3x^{3} dx = 3*4 = 12\).

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