Name: finding ellipse area with trig substitution
Uploaded: 2026-01-12T12:24:35.000Z
Duration: 1 min 23 s
Channel: JoVE

Molniya satellites, known for their highly elliptical orbits, trace distinct regions of space during each revolution around Earth.To estimate the area enclosed, the orbit is modeled as an ellipse in its standard form, which is then rearranged to an expression involving a square root term.Because the ellipse is symmetric about both axes, the area in the first quadrant is computed using integration and then multiplied by four to get the total area.Here, direct integration is difficult due to the square root. To resolve this, trigonometric substitution is used by substituting x in terms of the sine function. This substitution converts the square root term into a simple cosine function, and the differential changes accordingly.The integration limits are then updated to correspond to the angular domain, ranging from zero to pi over two. The integrand is rewritten using a trigonometric identity to simplify the integration process. This definite integral is evaluated over the new limits.The final expression yields the total area swept by the satellite during one complete orbit.

trigonometric substitution

Trigonometric substitution is a technique used to simplify integrals that contain square root expressions involving quadratic forms. It is particularly effective when the integrand includes terms resembling those found in standard geometric equations, such as circles or ellipses.Molniya satellites follow highly elliptical orbits, repeatedly sweeping out the same regions of space as they revolve around Earth. To estimate the area enclosed by such an orbit, the path is modeled as an ellipse written in standard form. This equation is then rearranged to isolate a square root expression, allowing the area to be computed using integration.Because an ellipse is symmetric about both the horizontal and vertical axes, the area in the first quadrant is calculated first and then multiplied by four to obtain the total area. However, direct integration of the resulting expression is difficult due to the square root term. Trigonometric substitution resolves this difficulty by expressing the variable xxx in terms of a sine function. This substitution transforms the square root into a cosine function, while the differential is rewritten accordingly.After substitution, the limits of integration are converted from Cartesian bounds to angular bounds, typically ranging from zero to &#960;/2. The integrand is then simplified using trigonometric identities, producing an expression that is straightforward to integrate. Evaluating the definite integral over the new limits yields the area in the first quadrant, and multiplying by four accounts for the symmetry of the ellipse.This approach provides an efficient and systematic method for computing the total area enclosed by an elliptical satellite orbit using trigonometric substitution.

Trigonometric Substitution

Trigonometric substitution is a technique used to simplify integrals that contain square root expressions involving quadratic forms. It is particularly ...

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JoVE Core

Calculus

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Techniques of Integration

KEY TERMS AND CONCEPTS

finding ellipse area with trig substitution

Trigonometric substitution is a method for solving integrals that contain square root expressions from quadratic forms. It works well when the integrand looks like a familiar geometric equation, such as a circle or an ellipse.

Here, the method is applied to estimate the area inside a Molniya satellite orbit. Molniya satellites move in highly elliptical orbits and repeatedly sweep across the same regions of space as they go around Earth. The orbit is modeled as an ellipse in standard form, and the equation is rearranged to isolate a square root term that can be handled by integration.

Because an ellipse is symmetric about both the horizontal and vertical axes, the area in the first quadrant is found first. That result is then multiplied by four to get the total area. The direct integral is still difficult because of the square root, so trigonometric substitution is used to simplify it.

The variable x is written in terms of a sine function. This changes the square root into a cosine expression and also changes the differential. The limits of integration are converted from Cartesian bounds to angular bounds, usually from zero to π/2. After that, trigonometric identities make the integrand easier to integrate. Evaluating the definite integral gives the first-quadrant area, and multiplying by four gives the total area enclosed by the elliptical orbit.

Finding Ellipse Area with Trig Substitution

finding-ellipse-area-with-trig-substitution

integration by parts: indefinite integrals

integration-by-parts-indefinite-integrals

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integrals-of-powers-of-secant-and-tangent

Solving Secant and Tangent Integrals

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Trigonometric substitution is a method for solving integrals that contain square root expressions from quadratic forms. It works well when the integrand looks like a familiar geometric equation, such as a circle or an ellipse. Here, the method is applied to estimate the area inside a Molniya satellite orbit. Molniya satellites move in highly elliptical orbits and repeatedly sweep across the same regions of space as they go around Earth. The orbit is modeled as an ellipse in standard form, and...

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integration-of-rational-functions-using-partial-fractions

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key terms and concepts

See how trigonometric substitution simplifies an ellipse-area integral for a Molniya satellite orbit by turning a square root expression into an easier trig form.

Finding Ellipse Area with Trig Substitution | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Finding Ellipse Area with Trig Substitution

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