Chapter 11: Q12.8P (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.
8. $\int_{0}^{\frac{\sqrt{3}}{2}} \frac{\sqrt{49 - 4 t^{2}}}{\sqrt{1 - t^{2}}} d t$

Short Answer

Expert verified

The value of integral in elliptic form is $7 E (\frac{\sqrt{3}}{2}, \frac{2}{7}) \approx 6.11662$ .

Step by step solution

Given Information

The value of integration is $\int_{0}^{\frac{\sqrt{3}}{2}} \frac{\sqrt{49 - 4 t^{2}}}{\sqrt{1 - t^{2}}} d t$ .

Definition of elliptic form

The elliptic form of the integral is defined as $E (1, k) = \int_{0}^{1} \frac{\sqrt{1 - k^{2} t^{2}}}{\sqrt{1 - t^{2}}} d t$ .

Find the value of Integral

The value of integration is $\int_{0}^{\frac{\sqrt{3}}{2}} \frac{\sqrt{49 - 4 t^{2}}}{\sqrt{1 - t^{2}}} d t$ .

Factor out 49 the equation becomes as follows,

$\begin{array}{rcl} l & = & \int_{0}^{\frac{\sqrt{3}}{2}} \frac{\sqrt{49 (1 - \frac{4 t^{2}}{49})}}{\sqrt{1 - t^{2}}} d t \\ = & 7 \int_{0}^{\frac{\sqrt{3}}{2}} \frac{\sqrt{(1 - \frac{4 t^{2}}{49})}}{\sqrt{1 - t^{2}}} d t \end{array}$

The formula for the beta function is $\begin{array}{rcl} E (1, K) & = & \int_{0}^{1} \frac{\sqrt{1 - k^{2} t^{2}}}{\sqrt{1 - t^{2}}} d t \end{array}$ .

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

$l = 7 \int_{0}^{\frac{\sqrt{3}}{2}} \frac{\sqrt{1 + \frac{2^{2} t^{2}}{7^{2}}}}{\sqrt{1 - t^{2}}} d t l = 7 E (\frac{\sqrt{3}}{2}, \frac{2}{7}) l \approx 6.11662$

Hence, the solution is mentioned below.

The value of integral in elliptic form is $\begin{array}{rcl} 7 E (\frac{\sqrt{3}}{2}, \frac{2}{7}) & \approx & 6.11662 \end{array}$ .

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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.
8. $\int_{0}^{\frac{\sqrt{3}}{2}} \frac{\sqrt{49 - 4 t^{2}}}{\sqrt{1 - t^{2}}} d t$

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

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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.8. ∫03249-4t21-t2dt

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

Thermodynamics

Oscillations

Waves Physics

Atoms and Radioactivity

Circular Motion and Gravitation

Conservation of Energy and Momentum

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.
8. $\int_{0}^{\frac{\sqrt{3}}{2}} \frac{\sqrt{49 - 4 t^{2}}}{\sqrt{1 - t^{2}}} d t$