Q. 79 Use Theorem 6.7 to prove that a ... [FREE SOLUTION]

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Chapter 6: Q. 79 (page 540)

Use Theorem 6.7 to prove that a circle of radius $r$ has circumference $2 蟺谤$ .

Short Answer

Expert verified

Circle of radius $r$ has circumference $2 蟺谤$ .

Step by step solution

Step 1. Given Information

Circle of radius $r$ .

Step 2. Proving circumference of circle of radius r to be 2πr

$Then the arc length of f (x) from x = a to x = b can be represented by the definite integral : {鈭�}_{a}^{b} \sqrt{1 + (f' {(x))}^{2}} d x Equation of circle of radius r i s : x^{2} + y^{2} = r^{2} Therefore, y = \sqrt{r^{2} - x^{2}} = f (x) and f' (x) = - \frac{x}{\sqrt{r^{2} - x^{2}}} Circumference of the circle is twice the arc length from x = - r to x = r of f (x) Therefore, C = 2 {鈭�}_{- r}^{r} \sqrt{1 + (f' {(x))}^{2}} d x C = 2 {鈭�}_{- r}^{r} \sqrt{1 + {(- \frac{x}{\sqrt{r^{2} - x^{2}}})}^{2}} d x C = 2 {鈭�}_{- r}^{r} \sqrt{1 + (\frac{x^{2}}{r^{2} - x^{2}})} d x C = 2 {鈭�}_{- r}^{r} \sqrt{{\frac{r^{2} - x^{2} + x}{r^{2} - x^{2}}}^{2}} d x C = 2 {鈭�}_{- r}^{r} \frac{r}{\sqrt{r^{2} - x^{2}}} d x Put, x = r \sin t, dx = r \cos t dt Limits of integration change to - \frac{蟺}{2} to \frac{蟺}{2} C = 2 {鈭�}_{- \frac{蟺}{2}}^{\frac{蟺}{2}} \frac{r 脳 r 脳 \cos t}{\sqrt{r^{2} - {(r \sin t)}^{2}}} dt C = 2 {鈭�}_{- \frac{蟺}{2}}^{\frac{蟺}{2}} \frac{r 脳 r 脳 \cos t}{\sqrt{r^{2} - {(r \sin t)}^{2}}} dt C = 2 r {鈭�}_{- \frac{蟺}{2}}^{\frac{蟺}{2}} \frac{r 脳 \cos t}{r 脳 \cos t} dt C = 2 r {鈭�}_{- \frac{蟺}{2}}^{\frac{蟺}{2}} 1 dt C = 2 r t_{- \frac{蟺}{2}}^{\frac{蟺}{2}} C = 2 r [\frac{蟺}{2} - (- \frac{蟺}{2})] C = 2 蟺谤$

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91影视

Use Theorem 6.7 to prove that a circle of radius $r$ has circumference $2 蟺谤$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving circumference of circle of radius r to be 2πr

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91影视

Use Theorem 6.7 to prove that a circle of radius rhas circumference2蟺谤.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Proving circumference of circle of radius r to be 2πr

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Pure Maths

Theoretical and Mathematical Physics

Logic and Functions

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

Use Theorem 6.7 to prove that a circle of radius $r$ has circumference $2 蟺谤$ .