Q. 11 Fill in the blanks to complete e... [FREE SOLUTION]

Chapter 5: Q. 11 (page 494)

Fill in the blanks to complete each of the following theorem statements:
If $a$ is a positive real number, and if $x 鈭�___$ and $u 鈭�_$ , then the substitution $x = a \sin u$ gives us $\sqrt{a^{2} - x^{2}} =___$ .

Short Answer

Expert verified

If $a$ is a positive real number, and if $x 鈭� [- a, a]$ and $u 鈭� [- \frac{蟺}{2}, \frac{蟺}{2}]$ , then the substitution $x = a \sin u$ gives us $\sqrt{a^{2} - x^{2}} = a \cos u$ .

Step by step solution

Step 1. Given information

If $a$ is a positive real number, and if role="math" localid="1653537491868" $x 鈭�_____$ and $u 鈭�_____$ , then the substitution $x = a \sin u$ gives us $\sqrt{a^{2} - x^{2}} =_____$ .

Step 2. Filling in the blanks to complete the theorem statements

If $a$ is a positive real number, and if $x 鈭� [- a, a]$ and $u 鈭� [- \frac{蟺}{2}, \frac{蟺}{2}]$ , then the substitution $x = a \sin u$ gives us $\sqrt{a^{2} - x^{2}} = a \cos u$ .

By substituting for $x$ and simplifying, we get the above answer valid under the conditions mentioned. That is, $\sqrt{a^{2} - x^{2}} = \sqrt{a^{2} - {(a \sin u)}^{2}}$

$= \sqrt{a^{2} - a^{2} \sin^{2} u} = \sqrt{a^{2} (1 - \sin^{2} u)} = \sqrt{a^{2} \cos^{2} u} = a \cos u$

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91影视

Fill in the blanks to complete each of the following theorem statements:
If $a$ is a positive real number, and if $x 鈭�___$ and $u 鈭�_$ , then the substitution $x = a \sin u$ gives us $\sqrt{a^{2} - x^{2}} =___$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Filling in the blanks to complete the theorem statements

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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91影视

Fill in the blanks to complete each of the following theorem statements: If ais a positive real number, and if x鈭�_____and u鈭�_____, then the substitution x=asinugives us a2-x2=_____.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Filling in the blanks to complete the theorem statements

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Mechanics Maths

Applied Mathematics

Decision Maths

Calculus

Logic and Functions

Study anywhere. Anytime. Across all devices.

Fill in the blanks to complete each of the following theorem statements:
If $a$ is a positive real number, and if $x 鈭�___$ and $u 鈭�_$ , then the substitution $x = a \sin u$ gives us $\sqrt{a^{2} - x^{2}} =___$ .