W02 - HW Solutions

04

Compute the definite integral:

Solution

Perform substitution . Draw right triangle with the length of the hypotenuse being 1 and that of the opposite being for reference.

Adjust the bounds:

Rewrite integral

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Simplify integral using Pythagorean identity

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Use trigonometric identity to rewrite integrand.

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Evaluate definite integral.

07

Compute the integral:

• (a) Using
• (b) Using

Solution (a)

Identify given .

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Perform -substitution

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Compute integral using power rule.

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Substitute back

(b)

Identify given

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Rewrite integrand as product of .

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Rewrite as using the Pythagorean trigonometric identity.

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Perform -substitution.

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Compute integral.

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Substitute back

08

Compute the integral:

Perform -substitution .

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Rewrite integrand as product of

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Perform substitution

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Compute integral

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Substitute back

12

Compute the definite integral:

Solution

Perform -substitution .

Adjust bounds:

Rewrite integral:

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Perform substitution . Draw a right triangle with the length of the hypotenuse being and opposite being for reference.

Adjust bounds:

Rewrite integral:

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Use Pythagorean identity to rewrite integrand.

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Multiply integrand by

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Use rule that . (If not immediately evident to you, use substitution , same thing).

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Evaluate expression.

13

A charged wire lies on the -axis running from to . The electric field at the point is given by:

Find the numerical value of assuming and and and .

Solution

Take constants outside of integral.

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Substitute given values inside integral: ,

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Use substitution . Draw right triangle with hypotenuse having length and opposite having length for reference.

Adjust bounds:

Rewrite integral

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Use Pythagorean identity to simplify integral.

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Evaluate integral. Plug back in given values for , , and .