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Calculus Practice: Volumes of Solids of Revolution (Cross Sections, Washer, Disk Methods)

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Q1. Suppose the region bounded by the function and the x-axis is revolved about the x-axis.

Background

Topic: Solids of Revolution (Disk Method)

This question tests your ability to set up and evaluate a definite integral for the volume of a solid formed by revolving a region around the x-axis.

Key Terms and Formulas:

  • Solid of Revolution: A 3D shape formed by rotating a 2D region about an axis.

  • Disk Method: Used when the solid has no hole; the volume is given by .

  • : The distance from the axis of rotation to the curve (radius).

Step-by-Step Guidance

  1. Identify the region: The curve and the x-axis. Find the x-values where to determine the bounds.

  2. Set up the integral: Since the region is revolved about the x-axis, use the disk method. The radius is .

  3. Write the definite integral: , where and are the x-intercepts.

  4. Expand the integrand and prepare to integrate, but do not compute the final value yet.

Solid of revolution about the x-axis

Try solving on your own before revealing the answer!

Final Answer:

After expanding and integrating, you would find the exact volume. The bounds are determined by solving .

Q2. Suppose the region (in Quadrant I) bounded by the function , the y-axis, and the line is revolved about the y-axis.

Background

Topic: Solids of Revolution (Washer Method)

This question tests your ability to set up a definite integral for the volume of a solid formed by revolving a region about the y-axis.

Key Terms and Formulas:

  • Washer Method: Used when the solid has a hole; the volume is .

  • : Outer radius from axis to outer curve.

  • : Inner radius from axis to inner curve.

Step-by-Step Guidance

  1. Express in terms of : Since , .

  2. Identify the bounds: ranges from $1x=0.

  3. Set up the integral: The region is between (y-axis) and .

  4. Write the definite integral for the volume using the shell or washer method, but do not evaluate.

Region bounded by $e^x$, y-axis, and $y=5$

Try solving on your own before revealing the answer!

Final Answer:

This integral represents the volume using the shell method, where goes from $0\ln 5$.

Q3. Suppose the region bounded by the function , the y-axis, and the line is revolved about the line $y = 5$.

Background

Topic: Solids of Revolution (Washer Method)

This question tests your ability to set up a definite integral for the volume of a solid formed by revolving a region about a horizontal line ().

Key Terms and Formulas:

  • Washer Method: .

  • : Distance from to the y-axis.

  • : Distance from to .

Step-by-Step Guidance

  1. Identify the bounds: goes from $0\ln 5e^x = 5$).

  2. Set up the radii: , .

  3. Write the definite integral: .

  4. Prepare to expand and integrate, but do not compute the final value yet.

Region revolved about y=5Solid of revolution about y=5

Try solving on your own before revealing the answer!

Final Answer:

This integral represents the volume using the washer method, with bounds from to .

Q4. Suppose the region bounded by the function and the lines and is revolved about the line $x = 2$.

Background

Topic: Solids of Revolution (Washer Method)

This question tests your ability to set up and evaluate a definite integral for the volume of a solid formed by revolving a region about a vertical line ().

Key Terms and Formulas:

  • Washer Method: .

  • : Distance from to the outer curve.

  • : Distance from to the inner curve.

Step-by-Step Guidance

  1. Express the region in terms of : .

  2. Identify the bounds: goes from the value where () up to .

  3. Set up the radii: (from to ), (from $x=2$ to ).

  4. Write the definite integral: .

  5. Expand the integrand and prepare to integrate, but do not compute the final value yet.

Problem 12.4 setupDesmos graph of region for Problem 12.4

Try solving on your own before revealing the answer!

Final Answer:

After expanding and integrating, you would find the exact volume. The bounds are determined by the intersection points.

Q5. Suppose the region bounded by the function and the function is revolved about the line .

Background

Topic: Solids of Revolution (Washer Method)

This question tests your ability to set up a definite integral for the volume of a solid formed by revolving a region about a horizontal line ().

Key Terms and Formulas:

  • Washer Method: .

  • : Distance from to the outer curve.

  • : Distance from to the inner curve.

Step-by-Step Guidance

  1. Identify the bounds: Find where to determine the limits of integration.

  2. Set up the radii: , .

  3. Write the definite integral: .

  4. Prepare to expand and integrate, but do not compute the final value yet.

Solid of revolution about y=4

Try solving on your own before revealing the answer!

Final Answer:

The bounds and are found by solving .

Q6. Suppose the region bounded by the function and the function is revolved about the line .

Background

Topic: Solids of Revolution (Washer Method)

This question tests your ability to set up a definite integral for the volume of a solid formed by revolving a region about a vertical line ().

Key Terms and Formulas:

  • Washer Method: .

  • : Distance from to the outer curve.

  • : Distance from to the inner curve.

Step-by-Step Guidance

  1. Express the region in terms of : and .

  2. Identify the bounds: Find where to determine the limits of integration.

  3. Set up the radii: , (where and are solved from and ).

  4. Write the definite integral: .

  5. Prepare to expand and integrate, but do not compute the final value yet.

Solid of revolution about x=-1Washer-shaped solid about x=-1

Try solving on your own before revealing the answer!

Final Answer:

The bounds and are found by solving .

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