Q1(a). Calculate the volume of the solid obtained by rotating the region from to about the -axis.

Background

Topic: Volumes of solids of revolution (Calculus II)

This question tests your ability to set up and compute the volume of a solid formed by rotating a region about the -axis, using integration.

Key Terms and Formulas

• Solid of revolution

• Disk/washer method

• Volume formula (about -axis):

Step-by-Step Guidance

1. Identify the bounds for . Since goes from $0, solve for when and .

2. Express in terms of : , so the region is bounded by and .

3. Set up the volume integral using the disk method about the -axis: .

4. Substitute into the formula: .

5. Determine the limits and by solving for when and .

Try solving on your own before revealing the answer!

Q1(b). Calculate the volume of the solid obtained by rotating the region from to about the -axis.

Background

Topic: Volumes of solids of revolution (washer method)

This question tests your ability to set up and compute the volume of a solid formed by rotating a region about the -axis.

Key Terms and Formulas

• Washer method

• Volume formula (about -axis): (for shell method), or (for disk/washer method)

Step-by-Step Guidance

1. Sketch the region: goes from $0, with from $0.

2. Since rotating about the -axis, consider using the shell method: , where is the height of the shell.

3. Here, .

4. Set up the integral: .

5. Expand and simplify the integrand before integrating.

Try solving on your own before revealing the answer!

Q1(c). Calculate the volume of the solid obtained by rotating the region from to about the -axis.

Background

Topic: Volumes of solids of revolution (disk method)

This question tests your ability to set up and compute the volume of a solid formed by rotating a region about the -axis.

Key Terms and Formulas

• Disk method

• Volume formula (about -axis):

Step-by-Step Guidance

1. Identify the bounds: goes from $0\frac{\pi}{4}$.

2. The region is bounded above by and below by .

3. Set up the volume integral: .

4. Simplify using trigonometric identities.

5. Prepare to integrate the resulting expression over .

Try solving on your own before revealing the answer!

Q1(d). Calculate the volume of the solid obtained by rotating the region from to about the -axis.

Background

Topic: Volumes of solids of revolution (shell method)

This question tests your ability to set up and compute the volume of a solid formed by rotating a region about the -axis.

Key Terms and Formulas

• Shell method

• Volume formula (about -axis):

Step-by-Step Guidance

1. Identify the bounds: goes from $0\sqrt{2}$.

2. For each , goes from $0y^2y^2$.

3. Set up the shell method integral: .

4. Simplify the integrand before integrating.

5. Prepare to integrate over .

Try solving on your own before revealing the answer!

Q1(e). Calculate the volume of the solid obtained by rotating the region for to about the -axis.

Background

Topic: Volumes of solids of revolution (disk method)

This question tests your ability to set up and compute the volume of a solid formed by rotating a region about the -axis.

Key Terms and Formulas

• Disk method

• Volume formula:

Step-by-Step Guidance

1. Identify the bounds: goes from $0\frac{\pi}{2}$.

2. The region is bounded above by and below by .

3. Set up the volume integral: .

4. Simplify to .

5. Prepare to integrate over .

Try solving on your own before revealing the answer!

Q1(f). Calculate the volume of the solid obtained by rotating the region from to about the -axis.

Background

Topic: Volumes of solids of revolution (shell method)

This question tests your ability to set up and compute the volume of a solid formed by rotating a region about the -axis.

Key Terms and Formulas

• Shell method

• Volume formula:

Step-by-Step Guidance

1. Identify the bounds: goes from $0.

2. For each , the height is .

3. Set up the shell method integral: .

4. Simplify the integrand: .

5. Prepare to integrate over .

Try solving on your own before revealing the answer!

Q2(a). Find the arclength of the curve from to .

Background

Topic: Arc length of a curve (Calculus II)

This question tests your ability to compute the length of a curve using the arc length formula.

Key Terms and Formulas

• Arc length formula:

Step-by-Step Guidance

1. Find for .

2. Compute and add $1$ to get the expression under the square root.

3. Set up the arc length integral: .

4. Simplify the integrand as much as possible.

5. Prepare to integrate over .

Try solving on your own before revealing the answer!

Q2(b). Find the arclength of the curve from to .

Background

Topic: Arc length of a curve (parametric form)

This question tests your ability to compute the length of a curve given as a function of .

Key Terms and Formulas

• Arc length formula:

Step-by-Step Guidance

1. Find for .

2. Compute and add $1$ to get the expression under the square root.

3. Set up the arc length integral: .

4. Simplify the integrand as much as possible.

5. Prepare to integrate over .

Try solving on your own before revealing the answer!

Q2(c). Find the arclength of the curve from to .

Background

Topic: Arc length of a curve (Calculus II)

This question tests your ability to compute the length of a curve using the arc length formula.

Key Terms and Formulas

• Arc length formula:

Step-by-Step Guidance

1. Find for .

2. Compute and add $1$ to get the expression under the square root.

3. Set up the arc length integral: .

4. Simplify the integrand as much as possible.

5. Prepare to integrate over .

Try solving on your own before revealing the answer!

Q3(a). Find the surface area of the surface of revolution , revolved around the -axis.

Background

Topic: Surface area of revolution (Calculus II)

This question tests your ability to compute the surface area of a solid formed by revolving a curve about the -axis.

Key Terms and Formulas

• Surface area formula:

Step-by-Step Guidance

1. Find for .

2. Compute and add $1$ to get the expression under the square root.

3. Set up the surface area integral: .

4. Simplify the integrand as much as possible.

5. Prepare to integrate over .

Try solving on your own before revealing the answer!

Q3(b). Find the surface area of the surface of revolution , revolved around the -axis.

Background

Topic: Surface area of revolution (Calculus II)

This question tests your ability to compute the surface area of a solid formed by revolving a curve about the -axis.

Key Terms and Formulas

• Surface area formula:

Step-by-Step Guidance

1. Find for .

2. Compute and add $1$ to get the expression under the square root.

3. Set up the surface area integral: .

4. Simplify the integrand as much as possible.

5. Prepare to integrate over .

Try solving on your own before revealing the answer!

Q3(c). Find the surface area of the surface of revolution , revolved around the -axis.

Background

Topic: Surface area of revolution (parametric form)

This question tests your ability to compute the surface area of a solid formed by revolving a curve about the -axis, given as a function of .

Key Terms and Formulas

• Surface area formula:

Step-by-Step Guidance

1. Find for .

2. Compute and add $1$ to get the expression under the square root.

3. Set up the surface area integral: .

4. Simplify the integrand as much as possible.

5. Prepare to integrate over .

Try solving on your own before revealing the answer!