Q1. Let . Compute .

Background

Topic: Double Integrals

This question tests your understanding of evaluating double integrals over a rectangular region. You need to integrate a function of two variables over the unit square.

Key Terms and Formulas

• Double Integral: computes the volume under over the region .

• Order of Integration: Integrate with respect to first, then (or vice versa, depending on the limits).

Step-by-Step Guidance

1. Write out the integral explicitly: .

2. Integrate with respect to first, treating as a constant: .

3. Compute the inner integral: .

4. Evaluate and substitute back into the outer integral.

Try solving on your own before revealing the answer!

Q2. The change of variables rule for double integrals is given by:

Background

Topic: Change of Variables in Multiple Integrals (Jacobian)

This question is about transforming the region of integration using a change of variables and the Jacobian determinant.

Key Terms and Formulas

• Change of Variables Formula:

• Jacobian Determinant:

Step-by-Step Guidance

1. Identify the transformation equations and .

2. Compute the Jacobian determinant for the transformation.

3. Rewrite the limits of integration in terms of and .

4. Set up the new double integral using the formula above.

Try solving on your own before revealing the answer!

Q3. The region is the area in the plane bounded by , , and . Compute .

Background

Topic: Double Integrals over Triangular Regions

This question tests your ability to set up and evaluate a double integral over a triangular region in the -plane.

Key Terms and Formulas

• Region : The triangle with vertices at , , and .

• Limits of Integration: For from $0, goes from $0.

Step-by-Step Guidance

1. Sketch or visualize the region to determine the correct limits for and .

2. Set up the double integral: .

3. Integrate with respect to first, treating as a constant.

4. Substitute the limits for and simplify the resulting expression before integrating with respect to .

Try solving on your own before revealing the answer!

Q4. Let . Compute , where is the straight line from to .

Background

Topic: Line Integrals of Vector Fields

This question tests your understanding of how to compute the line integral of a vector field along a straight path in the plane.

Key Terms and Formulas

• Line Integral:

• Parameterization: For a straight line from to , use for in .

Step-by-Step Guidance

1. Parameterize the curve as , .

2. Compute and .

3. Take the dot product: .

4. Set up the integral and prepare to evaluate.

Try solving on your own before revealing the answer!

Q5. Let . Compute , where is the unit circle traversed once counterclockwise.

Background

Topic: Line Integrals and Green's Theorem

This question tests your ability to compute a line integral around a closed curve, and possibly to use Green's Theorem to simplify the calculation.

Key Terms and Formulas

• Green's Theorem: for .

• Parameterization of the Unit Circle: , , .

Step-by-Step Guidance

1. Identify and for .

2. Compute and .

3. Set up the double integral over the unit disk using Green's Theorem.

4. Alternatively, parameterize the circle and set up the line integral directly.

Try solving on your own before revealing the answer!

Q6. The region is the triangle with vertices , , and . Compute .

Background

Topic: Double Integrals over Triangular Regions

This question is similar to Q3, but the integrand is instead of .

Key Terms and Formulas

• Region : The triangle with vertices at , , and .

• Limits of Integration: For from $0, goes from $0.

Step-by-Step Guidance

1. Set up the double integral: .

2. Integrate with respect to first, treating as a constant.

3. Substitute the limits for and simplify before integrating with respect to .

4. Prepare to evaluate the resulting single-variable integral.

Try solving on your own before revealing the answer!

Q7. The region is the triangle with vertices , , and . Compute using a change of variables , .

Background

Topic: Change of Variables in Double Integrals

This question tests your ability to use a change of variables (with the Jacobian) to evaluate a double integral over a triangular region.

Key Terms and Formulas

• Change of Variables Formula:

• Jacobian Determinant:

Step-by-Step Guidance

1. Express and in terms of and using the given transformations.

2. Compute the Jacobian determinant for the transformation.

3. Find the new region in the -plane corresponding to the original triangle .

4. Set up the new double integral in terms of and .

Try solving on your own before revealing the answer!