Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Calculus → Applications of Integration ↓


Volumes of Solids of Revolution


In calculus, using integration to find the volume of solids of revolution is a fascinating application. This process allows us to determine the volume of a three-dimensional object by rotating a two-dimensional sphere around an axis. This technique involves using definite integrals and has many applications in engineering, physics, and even daily life scenarios. In this explanation, we will learn how to calculate these volumes using simple language and visual-supported examples.

What is a solid of revolution?

A solid of revolution is a three-dimensional object formed when a two-dimensional figure is rotated around a straight line known as the axis of revolution. The simplest example of this is forming a cylinder by rotating a rectangle around one of its edges.

  Consider a rectangle with one side on the x-axis:
  ,
 ,
 ,
 ,

  Rotating this rectangle around the x-axis will form a cylinder.

Disk method

The disk method is used to find the volume of solids of revolution. It applies when the solid is rotated around the x-axis. The idea is to "cut" the solid into thin disk-shaped pieces and then add up their volumes using integration.

The formula for the disk method is as follows:

V = π ∫[a, b] [r(x)]² dx

Where:

  • V is the volume of the solid.
  • R(x) is the radius of the disc at position x.
  • [a, b] are the limits of integration.

Example: Finding volume using the disk method

Suppose we want to find the volume of the solid formed by rotating the region under the curve y = sqrt(x) about the x-axis from x = 0 to x = 4.

The function to integrate is:
V = π ∫[0, 4] (sqrt(x))² dx
  ∫[0, 4] x dx = π ∫ [0, 4] x dx
  = π [1/2 * x²] from 0 to 4
  = π (1/2 * 4² - 1/2 * 0²)
  = π (8)
  = 8π

The volume of the solid is 8π cubic units.

y = sqrt(x)

Washer method

Sometimes solids have a hole in the middle. For these we use the washer method. It is so named because each piece looks like a washer – a disk with a hole in the middle.

The formula for the washer method is:
V = π ∫[a, b] ([r(x)]² - [r(x)]²) dx

Where:

  • R(x) is the outer radius.
  • r(x) is the inner radius.
  • [a, b] are the limits of integration.

Example: Volume of a washer-shaped solid

Let us find the volume of the solid obtained by rotating the region between y = sqrt(x) and y = 1 about the x-axis from x = 1 to x = 4.

V = π ∫[1, 4] ([sqrt(x)]² - 1²) dx
  ∫[1, 4] (x - 1) dx = π
  = π [1/2 * x² - x] from 1 to 4
  = π (1/2 * 4² - 4 - (1/2 * 1² - 1))
  = π(8 – 4 – (0.5 – 1))
  = π(4 + 0.5)
  = 4.5π

The volume is 4.5π cubic units.

y = 1

Shell method

When the solid is rotated about the y-axis, the shell method is often more appropriate. This method considers a cylindrical shell rather than a disk or washer.

The formula for the Shell method is:

v = 2π ∫[a, b] (x * [f(x)]) dx

Where:

  • [f(x)] is the height of the cylindrical shell at position x.
  • [a, b] are the boundaries of the region.

Example: Finding volume using the shell method

Suppose we rotate the region under y = x² about the y-axis from x = 0 to x = 1.

V = 2π ∫[0, 1] (x * x²) dx
  = 2π ∫[0, 1] x³ dx
  = 2π [1/4 * x⁴] from 0 to 1
  = 2π (1/4 * 1 - 1/4 * 0)
  = 2π (1/4)
  = π/2

The volume is π/2 cubic units.

Applications in real life

Understanding the volume of solids of revolution isn't just for solving academic problems; it has many real-life applications. Engineers and architects use these methods to calculate the volume of materials, such as figuring out how much material is needed to manufacture machinery parts or build structures such as tunnels.

In the industrial field one can use these techniques to determine the capacity and dimensions of tanks and pipes. For example, using the shell method, one can calculate how much paint is needed to coat the inside of a cylindrical tank.

Summary and conclusion

Calculus provides us with powerful tools for understanding and calculating the volume of solids of revolution. The choice of method - whether it is the disk, washer or shell method - depends on the nature of the problem and the axis around which the figure rotates. By integrating the area under a curve and around an axis, we can quickly and accurately determine the volume of complex three-dimensional objects.

From simple cylinders to more complex shapes, these methods simplify calculations and provide definitive answers for otherwise complex geometries. Whether dealing with daily life problems or specialized engineering tasks, understanding how to calculate these volumes is an important skill.

Through this explanation, the steps and methods to calculate the volume of solids of revolution are covered with illustrative examples and visual aids. By practicing more examples, you will gain the confidence and skills to handle increasingly complex problems in calculus.


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