Chapter 11: Q12.11P (page 559)

In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.
11. $\int_{\frac{- π}{2}}^{\frac{3 π}{8}} \frac{1}{\sqrt{1 - \frac{9}{10} \sin^{2} θ}} d θ$

Short Answer

Expert verified

The value of integral in elliptic form is l = 4.0968.

Step by step solution

Given Information

The given integral is $\int_{\frac{- π}{2}}^{\frac{3 π}{8}} \frac{1}{\sqrt{1 - \frac{9}{10} \sin^{2} θ}} d θ$ .

Definition of elliptic form

The elliptic form of the integral is defined as $F (\frac{π}{2}, k) = \int_{0}^{\frac{π}{2}} \frac{1}{\sqrt{1 - k^{2} s i n^{2} θ}} d θ$ .

Find the value of Integral

Let the given integral is $\int_{\frac{- π}{2}}^{\frac{3 π}{8}} \frac{1}{\sqrt{1 - \frac{9}{10} \sin^{2} θ}} d θ$ .

Rewrite the integral, equation as follows,

$l = \int_{\frac{- π}{2}}^{0} \frac{1}{\sqrt{1 - \frac{9}{10} \sin^{2} θ}} d θ + \int_{0}^{\frac{3 π}{8}} \frac{1}{\sqrt{1 - \frac{9}{10} \sin^{2} θ}} d θ$

The formula for the beta function is $F (\frac{π}{2}, K) = \int_{0}^{\frac{π}{2}} \frac{1}{\sqrt{1 - k^{2} \sin^{2} θ}} d θ$ .

Equate the above equation with the value of I, the value of I becomes follows.

$l = \int_{\frac{- π}{2}}^{0} \frac{1}{\sqrt{1 - \frac{9}{10} \sin^{2} θ}} d θ + \int_{0}^{\frac{3 π}{8}} \frac{1}{\sqrt{1 - \frac{9}{10} \sin^{2} θ}} d θ l = F (\frac{3 π}{8}, \frac{3}{\sqrt{10}}) + F (\frac{π}{2}, \frac{3}{\sqrt{10}})$

On simplifying further, we get,

$\begin{array}{rcl} l & \approx & 1.51871 + 2.57809 \\ = & 4.0968 \end{array}$

The value of integral in elliptic form is $\begin{array}{rcl} l & = & 4.0968 \end{array}$ .

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In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.
11. $\int_{\frac{- π}{2}}^{\frac{3 π}{8}} \frac{1}{\sqrt{1 - \frac{9}{10} \sin^{2} θ}} d θ$

Short Answer

Step by step solution

Given Information

Definition of elliptic form

Find the value of Integral

One App. One Place for Learning.

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In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.11. ∫-π23π811-910sin2θdθ

Short Answer

Step by step solution

Given Information

Definition of elliptic form

Find the value of Integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Translational Dynamics

Fluids

Measurements

Physics of Motion

Astrophysics

Turning Points in Physics

Study anywhere. Anytime. Across all devices.

In Problem 4 to 13, identify each of the integral as an elliptic (see Example 1 and 2). Learn the notation of your computer program (see Problem 3) and then evaluate the integral by computer.
11. $\int_{\frac{- π}{2}}^{\frac{3 π}{8}} \frac{1}{\sqrt{1 - \frac{9}{10} \sin^{2} θ}} d θ$