Chapter 9: Q. 14. (page 755)

Give a geometric explanation of why
$\int_{0}^{\frac{2 π}{n}} r^{2} d θ = π r^{2}$
for any positive real number r and any positive integer $n$ Would the equation also hold for non-integer values of $n$ ?

Short Answer

Expert verified

The equation $\int_{0}^{\frac{2 π}{n}} r^{2} d θ = π r^{2}$ holds true for any non-integer values of $n$

Step by step solution

Given information

Now consider integral $\int_{0}^{\frac{2 π}{n}} r^{2} d θ$

The objective is to find to prove that the integral is equal to πr2

$π r^{2}$ for non-integer values of $n$

Take the integral now.

$\frac{n}{2} \int_{0}^{\frac{2 π}{n}} r^{2} d θ = \frac{n r^{2}}{2} \int_{0}^{\frac{2 π}{n}} 1 d θ$

Prove that any positive real number r and any positive integer Would the equation also hold for non-integer values of n?

If $r$ is a positive real number and $n$ is a non-negative integer, then That is, both $n$ and $r$ are constants. As a result, they are removed, and the remaining value is integrated within the stated limitations.

$\int_{0}^{\frac{2 π}{n}} r^{2} d θ = \frac{n r^{2}}{2} {[θ]}^{2} = \frac{n r^{2}}{2} [\frac{2 π}{n} - 0] = π r^{2}$

Hence, the equation holds true for any non-integer values of $n$

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Give a geometric explanation of why
$\int_{0}^{\frac{2 π}{n}} r^{2} d θ = π r^{2}$
for any positive real number r and any positive integer $n$ Would the equation also hold for non-integer values of $n$ ?

Short Answer

Step by step solution

Given information

The objective is to find to prove that the integral is equal to πr2

Prove that any positive real number r and any positive integer Would the equation also hold for non-integer values of n?

One App. One Place for Learning.

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Give a geometric explanation of why∫02πnr2dθ=πr2for any positive real number r and any positive integer nWould the equation also hold for non-integer values of n?

Short Answer

Step by step solution

Given information

The objective is to find to prove that the integral is equal to πr2

Prove that any positive real number r and any positive integer Would the equation also hold for non-integer values of n?

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

Give a geometric explanation of why
$\int_{0}^{\frac{2 π}{n}} r^{2} d θ = π r^{2}$
for any positive real number r and any positive integer $n$ Would the equation also hold for non-integer values of $n$ ?