Chapter 11: Q12.14P (page 559)

Find the circumference of the ellipse $4 x^{2} + 9 y^{2} = 36$ .

Short Answer

Expert verified

The circumference of ellipse is $12 E (\frac{π}{2}, \frac{\sqrt{5}}{3}) \approx 1.32211$ .

Step by step solution

Given Information

The equation of ellipse is $4 x^{2} + 9 y^{2} = 36$ .

Definition of the circumference of ellipse.

The circumference of ellipse defined as $4 a \int_{0}^{\frac{π}{2}} \sqrt{1 - \frac{a^{2} - b^{2}}{a^{2}} s i n^{2} θ} d θ$ .

Find the value of Integral.

The equation of ellipse is $4 x^{2} + 9 y^{2} = 36$ .

Substitute the values given below in the equation mentioned above.

a² = 9

b² = 4

The circumference of ellipse defined as $C = 4 a \int_{0}^{\frac{π}{2}} \sqrt{1 - \frac{a^{2} - b^{2}}{a^{2}} \sin^{2} θ} d θ$ .

The equation becomes as follows.

$C = 4 a \int_{0}^{\frac{π}{2}} \sqrt{1 - \frac{a^{2} - b^{2}}{a^{2}} \sin^{2} θ} d θ C = 4 \times 2 \int_{0}^{\frac{π}{2}} \sqrt{1 - \frac{9 - 4}{9} \sin^{2} θ} d θ C = 4 \times 2 \int_{0}^{\frac{π}{2}} \sqrt{1 - \frac{5}{4} \sin^{2} θ} d θ$

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

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

$C = 4 \times 2 \int_{0}^{\frac{π}{2}} \sqrt{1 - \frac{5}{4} \sin^{2} θ} d θ C = 12 E (\frac{π}{2}, \frac{\sqrt{5}}{3}) C \approx 1.32211$

Hence, the solution is mentioned below.

The circumference of ellipse is $12 E (\frac{π}{2}, \frac{\sqrt{5}}{3}) \approx 1.32211$ .

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Find the circumference of the ellipse $4 x^{2} + 9 y^{2} = 36$ .

Short Answer

Step by step solution

Given Information

Definition of the circumference of ellipse.

Find the value of Integral.

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Find the circumference of the ellipse4x2+9y2=36.

Short Answer

Step by step solution

Given Information

Definition of the circumference of ellipse.

Find the value of Integral.

One App. One Place for Learning.

Most popular questions from this chapter

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Find the circumference of the ellipse $4 x^{2} + 9 y^{2} = 36$ .