Mass Calculation in a Bounded Region

The video tutorial explains how to calculate the mass of a region bounded by the function y = cosine X and the x-axis over the interval from 0 to Pi/2, using a density function. The process involves setting up a double integral with appropriate limits, performing integration with respect to y and x, and using integration by parts for complex integrals. The final result is obtained by evaluating the integral and simplifying the expression.

What is the function that bounds the region from above?

• y = sine X

• y = X squared

• y = cosine X

• y = tangent X

What is the density function used in the problem?

• rho(x, y) = x + y

• rho(x, y) = xy^2

• rho(x, y) = 2xy

• rho(x, y) = x^2 + y^2

What are the limits of integration for y?

• 0 to Pi

• 0 to Pi/2

• 0 to cosine X

• 0 to sine X

What is the anti-derivative of 2xy with respect to y?

• 2x^2y

• x^2y

• xy^2

• 2xy^2

Which method is used to solve the second integral with respect to x?

• Integration by substitution

• Integration by parts

• Partial fraction decomposition

• Trigonometric substitution

What substitution is used for integrating cosine 2x?

• u = x

• u = 2x

• u = cosine x

• u = sine x

What is the final expression for the mass of the region?

• Pi^2 / 8 - 1/2

• Pi^2 / 16 - 1/4

• Pi^2 / 2 - 1/16

• Pi^2 / 4 - 1/8

What is the approximate value of the mass calculated?

• 0.3669

• 0.25

• 0.75

• 0.5

What is the value of cosine Pi?

• 0.5

• -1

• 1

• 0

What is the value of sine Pi?

• 0.5

• -1

• 0

• 1