Understanding Double Integrals and Change of Variables

The video tutorial explains how to evaluate a double integral over a region defined by an ellipse. It demonstrates transforming the region into a circle using a change of variables, calculating the Jacobian determinant, and simplifying the integration process by converting to polar coordinates. The tutorial concludes with evaluating the integral and providing the final result.

What is the shape of the region defined by the inequality 25x² + 4y² ≤ 100?

• Parabola

• Ellipse

• Rectangle

• Circle

What transformation is applied to convert the elliptical region into a circular one?

• x = 5u, y = 2v

• x = 4u, y = 25v

• x = 2u, y = 5v

• x = u/2, y = v/5

What is the purpose of the Jacobian in the change of variables?

• To convert the integral into a single variable

• To adjust the integrand for the change of variables

• To transform the region into a square

• To find the limits of integration

How is the double integral simplified using polar coordinates?

• By converting the region into a square

• By changing the limits of integration

• By introducing a new variable

• By using trigonometric identities

What are the limits of integration for r in polar coordinates?

• 0 to 2π

• 0 to 10

• 0 to 5

• 0 to 1

What is the antiderivative of 40r³cos²θ with respect to r?

• 5r⁴cosθ

• 20r³cosθ

• 40r²cos²θ

• 10r⁴cos²θ

Which trigonometric identity is used to simplify the integration of cos²θ?

• cos²θ = 1 - sin²θ

• cos²θ = 1/2(1 + cos2θ)

• cos²θ = sin²θ + cos²θ

• cos²θ = 1/2(1 - cos2θ)

What is the final result of the evaluated integral?

• 15π

• 20π

• 5π

• 10π

What substitution is used for integrating 5cos2θ?

• u = θ, du = dθ

• u = 2θ, du = 2dθ

• u = sinθ, du = cosθdθ

• u = cosθ, du = -sinθdθ

What is the approximate numerical value of the final result?

• 31.1416

• 28.2743

• 15.7079

• 62.8318