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: 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 sketching.

3. Set up the integral for the area: .

4. Simplify the integrand and prepare to integrate.

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 of intersection and set up the appropriate integrals.

Key Terms and Formulas:

• Area between curves:

• Intersection points: Solve and for .

Step-by-Step Guidance

1. Find where to determine the intersection points.

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

3. Sketch or analyze which curve is above the other between the intersection points.

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

Try solving on your own before revealing the answer!

Q3. Sketch the graphs of , , , and .

Background

Topic: Polar Coordinates and Graphing

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

Key Terms and Formulas:

• Polar coordinates:

• Equation forms: (circle), (line), (vertical line), (manipulate to standard form)

Step-by-Step Guidance

1. For , recognize this is a straight line at that angle from the origin.

2. For , this is a circle of radius 5 centered at the origin.

3. For , rewrite as (vertical line in Cartesian coordinates).

4. For , divide both sides by (assuming ) to get , which is in Cartesian coordinates.

Try sketching these on your own before checking 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 in polar coordinates:

• Cardioid: A heart-shaped curve defined by

Step-by-Step Guidance

1. Determine the range of that traces the entire cardioid (usually $0).

2. Set up the area integral: .

3. Expand to simplify the integrand.

4. Prepare to integrate term by term.

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 asks you to find the area of a single petal of a rose curve using polar integration.

Key Terms and Formulas:

• Area in polar coordinates:

• Rose curve: has four petals.

Step-by-Step Guidance

1. Find the range of that traces one petal (e.g., $0\frac{\pi}{2}$).

2. Set up the area integral: .

3. Use the 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 within a circle in polar coordinates and bounded by a vertical line, requiring careful setup of limits.

Key Terms and Formulas:

• Area in polar coordinates:

• is a circle; is a vertical line at .

Step-by-Step Guidance

1. Find the intersection points between and to determine the limits of .

2. Set up the area integral for the region to the right of and inside .

3. Express the area as .

4. Simplify the integrand and prepare to integrate.

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 square cross-sections perpendicular to the y-axis.

Key Terms and Formulas:

• Volume by cross-sections: where is the area of the cross-section at .

• For squares:

Step-by-Step Guidance

1. Find the limits for by solving and for where the region exists.

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

3. Set up the area of the square cross-section: .

4. Set up the volume integral: .

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 asks you to find the volume of a solid with semicircular cross-sections perpendicular to the x-axis.

Key Terms and Formulas:

• Volume by cross-sections:

• Area of a semicircle:

Step-by-Step Guidance

1. Find the limits for by solving and for where the region exists.

2. For each , the diameter of the semicircle is the distance between the two curves: .

3. The radius is half the diameter.

4. Set up the area of the semicircular cross-section: .

5. 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 perpendicular to the y-axis.

Key Terms and Formulas:

• Volume by cross-sections:

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

Step-by-Step Guidance

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

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

3. Set up the area of the equilateral triangle cross-section: .

4. Set up the volume integral: .

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: or

Step-by-Step Guidance

1. Find the intersection points for by setting .

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

3. Express the radii and heights in terms of or as needed for each part.

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

Try setting up the integrals on your own before checking 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 the intersection points for by setting .

2. For each axis of rotation, determine the appropriate method and set up the integral.

3. Express the radii and heights in terms of or as needed for each part.

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

Try setting up the integrals on your own before checking 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 the intersection points for by setting and .

2. For each axis of rotation, determine the appropriate method and set up the integral.

3. Express the radii and heights in terms of or as needed for each part.

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

Try setting up the integrals on your own before checking 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 of integration ( from $0\pi/4$).

2. For each axis of rotation, determine the appropriate method and set up the integral.

3. Express the radii and heights in terms of or as needed for each part.

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

Try setting up the integrals on your own before checking 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 of integration ( from $1e$).

2. For each , the radius is the distance from to .

3. The height is .

4. Set up the shell method integral: .

Try setting up the integral on your own before checking 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 using calculus.

Key Terms and Formulas:

• Arc length formula:

Step-by-Step Guidance

1. Find for .

2. Square and add 1 to get 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 using calculus.

Key Terms and Formulas:

• Arc length formula:

Step-by-Step Guidance

1. Find for .

2. Square and add 1 to get 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 using calculus.

Key Terms and Formulas:

• Arc length formula:

Step-by-Step Guidance

1. Find for .

2. Square and add 1 to get 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!