Chapter 6: Q. 79 (page 540)

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

Short Answer

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

Step by step solution

Step 1. Given Information

Circle of radius $r$ .

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

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

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

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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Use Theorem 6.7 to prove that a circle of radius rhas circumference2πr.

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

Mechanics Maths

Applied Mathematics

Statistics

Geometry

Logic and Functions

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

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