Q1. Find the area enclosed by the graphs of and .

Background

Topic: Area Between Curves

This question tests your ability to find the area between two curves by integrating the difference of their functions over the interval where they intersect.

Key Terms and Formulas:

• Area between curves formula: where is the upper curve and is the lower curve.

• Intersection points: Solve to find the limits of integration.

Step-by-Step Guidance

1. Set and solve for to find the intersection points.

2. Determine which function is on top (greater) between the intersection points by comparing values or analyzing the graphs.

3. Set up the integral for the area: .

4. Simplify the integrand before integrating.

Try solving on your own before revealing the answer!

Q2. Find the area of the region bounded by , , and the x-axis.

Background

Topic: Area Between Curves and the x-axis

This question asks you to find the area between two parabolas and the x-axis, requiring you to determine the points where the curves intersect and where they meet the x-axis.

Key Terms and Formulas:

• Area formula:

• Intersection points: Solve and for .

Step-by-Step Guidance

1. Find the intersection points between and .

2. Find where each curve meets the x-axis ().

3. Determine the relevant intervals for integration.

4. Set up the integral(s) for the area, using the appropriate limits and functions.

Try solving on your own before revealing the answer!

Q3. Sketch the graphs of , , , and .

Background

Topic: Polar Coordinates and Graphs

This question tests your understanding of polar equations and how to sketch their graphs.

Key Terms and Formulas:

• Polar coordinates:

• Equation types: lines (, ), circles, and other curves.

Step-by-Step Guidance

1. Interpret each equation: is a straight line at that angle; is a circle of radius 5.

2. Rewrite in Cartesian form: .

3. Rewrite as or in Cartesian coordinates.

4. Sketch each graph on polar axes, labeling each curve.

Try sketching on your own before revealing the answer!

Q4. Find the area of the region enclosed by the cardioid .

Background

Topic: Area in Polar Coordinates

This question tests your ability to compute the area enclosed by a polar curve using integration.

Key Terms and Formulas:

• Area formula in polar coordinates:

• Limits: For a cardioid, typically goes from $0.

Step-by-Step Guidance

1. Identify the limits for (usually $0 for a full cardioid).

2. Set up the area integral: .

3. Expand to simplify the integrand.

4. Prepare to integrate each term separately.

Try solving on your own before revealing the answer!

Q5. Find the area of one leaf of the four-petaled rose .

Background

Topic: Area in Polar Coordinates

This question tests your ability to find the area of a single petal of a polar curve (rose curve).

Key Terms and Formulas:

• Area formula:

• Limits: For one petal, find the interval for that traces out one leaf.

Step-by-Step Guidance

1. Determine the interval for that corresponds to one leaf (e.g., $0\frac{\pi}{2}$).

2. Set up the area integral: .

3. Use a double-angle identity to simplify .

4. Prepare to integrate the simplified expression.

Try solving on your own before revealing the answer!

Q6. Find the area inside the circle and to the right of the vertical line .

Background

Topic: Area in Polar Coordinates

This question involves finding the area inside a circle in polar coordinates and bounded by a vertical line, requiring careful determination of the limits for .

Key Terms and Formulas:

• Area formula:

• Limits: Find where intersects to determine bounds.

Step-by-Step Guidance

1. Set and solve for to find intersection points.

2. Determine the interval for corresponding to the region to the right of the line.

3. Set up the area integral using the correct limits and .

4. Simplify the integrand if possible before integrating.

Try solving on your own before revealing the answer!

Q7. Find the volume of the solid whose base is the region bounded by and and whose cross-sections are squares with bases perpendicular to the y-axis.

Background

Topic: Volumes by Cross-Sections

This question tests your ability to find the volume of a solid with a given base and cross-sectional shape (squares), using integration.

Key Terms and Formulas:

• Volume formula:

• For squares: area = (side length)

Step-by-Step Guidance

1. Find the limits for by solving and for intersection points.

2. For each , the base of the square is the distance between the two curves: .

3. Set up the volume integral: .

4. Simplify the integrand before integrating.

Try solving on your own before revealing the answer!

Q8. Find the volume of the solid whose base is the region bounded by and and whose cross-sections are semicircles perpendicular to the x-axis.

Background

Topic: Volumes by Cross-Sections

This question tests your ability to find the volume of a solid with semicircular cross-sections, using integration.

Key Terms and Formulas:

• Volume formula:

• Area of semicircle:

• Diameter is the distance between the curves.

Step-by-Step Guidance

1. Find the limits for by solving and for intersection points.

2. For each , the diameter is .

3. Radius is half the diameter.

4. Set up the volume integral: .

Try solving on your own before revealing the answer!

Q9. Find the volume of the solid whose base is the region bounded by and and whose cross-sections are equilateral triangles with the base perpendicular to the y-axis.

Background

Topic: Volumes by Cross-Sections

This question tests your ability to find the volume of a solid with equilateral triangle cross-sections, using integration.

Key Terms and Formulas:

• Volume formula:

• Area of equilateral triangle: where is the side length.

Step-by-Step Guidance

1. Find the limits for by solving and for .

2. For each , the base of the triangle is the distance between the two values for that .

3. Set up the volume integral: .

4. Simplify the integrand before integrating.

Try solving on your own before revealing the answer!

Q10. Consider the region bounded by and . Set up, but do not solve, an integral to find the volume of the solid generated by revolving this region around:

• a) the x-axis.

• b) the line .

• c) the line .

Background

Topic: Volumes of Revolution (Washer/Disk/Shell Method)

This question tests your ability to set up integrals for volumes of solids of revolution using different axes.

Key Terms and Formulas:

• Washer method:

• Shell method:

Step-by-Step Guidance

1. Find the intersection points for to determine limits.

2. For each axis, identify the appropriate method (washer or shell) and set up the integral.

3. Express the radii or heights in terms of or as needed.

4. Write the integral setup for each part, but do not solve.

Try setting up the integrals on your own before revealing the answer!

Q11. Consider the region bounded by and . Find the volume of the solid generated by rotating this region around:

• a) the y-axis.

• b) the line (Just set up the integral).

• c) the line (Just set up the integral).

Background

Topic: Volumes of Revolution (Washer/Shell Method)

This question tests your ability to set up and compute volumes of solids of revolution for regions bounded by curves.

Key Terms and Formulas:

• Washer method:

• Shell method:

Step-by-Step Guidance

1. Find intersection points for to determine limits.

2. For each axis, identify the method and set up the integral.

3. Express radii or heights in terms of or as needed.

4. Write the integral setup for each part, but do not solve.

Try setting up the integrals on your own before revealing the answer!

Q12. Consider the region bounded by , , and in the first quadrant. Find the volume of the solid generated by revolving this region around:

• a) the y-axis.

• b) the line (Just set up the integral).

• c) the line (Just set up the integral).

Background

Topic: Volumes of Revolution (Washer/Shell Method)

This question tests your ability to set up and compute volumes of solids of revolution for regions bounded by curves.

Key Terms and Formulas:

• Washer method:

• Shell method:

Step-by-Step Guidance

1. Find intersection points for to determine limits.

2. For each axis, identify the method and set up the integral.

3. Express radii or heights in terms of or as needed.

4. Write the integral setup for each part, but do not solve.

Try setting up the integrals on your own before revealing the answer!

Q13. Consider the region bounded by on . Find the volume of the solid generated by rotating this region around:

• a) the x-axis.

• b) the line (Just set up the integral).

• c) the line (Just set up the integral).

Background

Topic: Volumes of Revolution (Washer/Shell Method)

This question tests your ability to set up and compute volumes of solids of revolution for regions bounded by a function.

Key Terms and Formulas:

• Washer method:

• Shell method:

Step-by-Step Guidance

1. Identify the limits for ($0\pi/4$).

2. For each axis, identify the method and set up the integral.

3. Express radii or heights in terms of or as needed.

4. Write the integral setup for each part, but do not solve.

Try setting up the integrals on your own before revealing the answer!

Q14. Consider the region bounded by , , and . Find the volume of the solid formed by revolving this region about the line .

Background

Topic: Volumes of Revolution (Shell Method)

This question tests your ability to set up and compute the volume of a solid formed by revolving a region about a vertical line not on the axis.

Key Terms and Formulas:

• Shell method:

Step-by-Step Guidance

1. Identify the limits for ($1e$).

2. For each , the radius is .

3. The height is .

4. Set up the shell integral: .

Try setting up the integral on your own before revealing the answer!

Q15. Find the arc length of over .

Background

Topic: Arc Length of a Curve

This question tests your ability to compute the arc length of a function over a given interval.

Key Terms and Formulas:

• Arc length formula:

Step-by-Step Guidance

1. Find for .

2. Square and add 1 to form the integrand.

3. Set up the arc length integral: .

4. Simplify the integrand as much as possible before integrating.

Try solving on your own before revealing the answer!

Q16. Find the arc length of over .

Background

Topic: Arc Length of a Curve

This question tests your ability to compute the arc length of a function over a given interval.

Key Terms and Formulas:

• Arc length formula:

Step-by-Step Guidance

1. Find for .

2. Square and add 1 to form the integrand.

3. Set up the arc length integral: .

4. Simplify the integrand as much as possible before integrating.

Try solving on your own before revealing the answer!

Q17. Find the arc length of over .

Background

Topic: Arc Length of a Curve

This question tests your ability to compute the arc length of a function over a given interval.

Key Terms and Formulas:

• Arc length formula:

Step-by-Step Guidance

1. Find for .

2. Square and add 1 to form the integrand.

3. Set up the arc length integral: .

4. Simplify the integrand as much as possible before integrating.

Try solving on your own before revealing the answer!