Area Under Curves and the Trapezoidal Rule for SS 3

Class: SS 3
Term: First Term
Week: 4
Age: 16-17 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Calculus: Integration
Previous Lesson: Definite Integrals and Applications to Kinematics
Topic: INTEGRATION
Subject Matter: Area under the curve; Trapezoidal rule; Volume of solids of revolution

Specific Objectives

By the end of the lesson, pupils/students should be able to:

Cognitive Domain

• Define the area under a curve.
• State the formula for calculating the area under a curve.
• Explain the trapezoidal rule for approximating area.
• State the formula for the trapezoidal rule.
• State the formulas for the volume of solids of revolution about the x-axis and y-axis.
• Calculate the area under a curve using definite integration.
• Approximate the area under a curve using the trapezoidal rule.
• Calculate the volume of solids of revolution.

Psychomotor Domain

• Sketch curves to determine the limits of integration.
• Apply the trapezoidal rule to solve practical problems.
• Solve problems involving areas bounded by two curves.
• Demonstrate the steps for calculating the volume of solids of revolution.

Affective Domain

• Appreciate the practical applications of integration in real-world scenarios.
• Develop accuracy and precision in mathematical calculations.
• Show interest in solving complex integration problems.

Reference Materials

The following resources were used in planning this lesson:

• 2025 Revised 9 Years Basic Education Curriculum
• Relevant State Unified Scheme of Work
• Further Mathematics for Senior Secondary Schools (A suitable textbook)
• The HeadTeacher Scheme of work

Instructional Materials

The teacher will teach this lesson with the aid of:

• Charts showing integration concepts, including areas bounded by curves.
• Whiteboard and markers.
• Calculators.
• Graph paper.

Rationale for the Lesson

This lesson is important as it equips students with advanced tools in calculus for solving real-world problems. Understanding area under curves and volumes of revolution is fundamental in fields like engineering, physics, and economics. The trapezoidal rule provides a practical method for approximating areas when exact integration is difficult or impossible.

Prerequisite/Previous Knowledge

Students should have prior knowledge of basic integration techniques, differentiation, algebraic manipulation, and sketching graphs of functions.

Lesson Content/Board Summary

INTEGRATION

Area Under a Curve

The area under a curve represents the region bounded by the function’s graph, the x-axis (or y-axis), and specific vertical (or horizontal) lines. Definite integration is used to find the exact area.

1. Area bounded by (y = f(x)), the x-axis, and lines (x=a) and (x=b):

( text{Area} = int_{a}^{b} y , dx = int_{a}^{b} f(x) , dx )

2. Area bounded by (x = g(y)), the y-axis, and lines (y=c) and (y=d):

( text{Area} = int_{c}^{d} x , dy = int_{c}^{d} g(y) , dy )

3. Area bounded by two curves (y_1 = f(x)) and (y_2 = g(x)):

If (f(x) ge g(x)) over the interval ([a, b]), then:

( text{Area} = int_{a}^{b} (y_1 – y_2) , dx = int_{a}^{b} (f(x) – g(x)) , dx )

Example 1

Question: Find the area bounded by the curve (y = x^2), the x-axis, and the lines (x=1) and (x=3).

Solution:

Step 1: Write the formula for area under the curve.

( text{Area} = int_{a}^{b} f(x) , dx )

Step 2: Substitute the function and limits.

( text{Area} = int_{1}^{3} x^2 , dx )

Step 3: Integrate the function.

( text{Area} = left[ frac{x^3}{3} right]_{1}^{3} )

Step 4: Evaluate at the limits.

( text{Area} = left( frac{3^3}{3} right) – left( frac{1^3}{3} right) = frac{27}{3} – frac{1}{3} = 9 – frac{1}{3} = frac{27-1}{3} = frac{26}{3} )

Answer: ( frac{26}{3} ) square units.

Trapezoidal Rule

The trapezoidal rule is a numerical method for approximating the definite integral of a function. It approximates the area under the curve by dividing the region into a series of trapezoids.

Formula

( int_{a}^{b} f(x) , dx approx frac{h}{2} [ (y_0 + y_n) + 2(y_1 + y_2 + dots + y_{n-1}) ] )

Where:

1. (h) = width of each strip (interval) = ( frac{b-a}{n} )
2. (n) = number of strips (intervals)
3. (a) = lower limit of integration
4. (b) = upper limit of integration
5. (y_0, y_1, dots, y_n) are the corresponding function values ((f(x))) at (x_0, x_1, dots, x_n).

Example 1

Question: Use the trapezoidal rule with 4 strips to estimate the area under the curve (y = x^2) from (x=0) to (x=4).

Solution:

Step 1: Determine (h).

( h = frac{b-a}{n} = frac{4-0}{4} = 1 )

Step 2: Find the x-values and corresponding y-values.

The x-values are (x_0=0, x_1=1, x_2=2, x_3=3, x_4=4).

The y-values are:

1. (y_0 = f(0) = 0^2 = 0)
2. (y_1 = f(1) = 1^2 = 1)
3. (y_2 = f(2) = 2^2 = 4)
4. (y_3 = f(3) = 3^2 = 9)
5. (y_4 = f(4) = 4^2 = 16)

Step 3: Apply the trapezoidal rule formula.

( text{Area} approx frac{h}{2} [ (y_0 + y_n) + 2(y_1 + y_2 + dots + y_{n-1}) ] )

( text{Area} approx frac{1}{2} [ (0 + 16) + 2(1 + 4 + 9) ] )

Step 4: Simplify and calculate.

( text{Area} approx frac{1}{2} [ 16 + 2(14) ] = frac{1}{2} [ 16 + 28 ] = frac{1}{2} [ 44 ] = 22 )

Answer: Approximately 22 square units.

Volume of Solids of Revolution

A solid of revolution is a three-dimensional shape formed by rotating a two-dimensional curve around an axis. The volume of such a solid can be calculated using integration.

Formulas

1. Revolution about the x-axis:

If the region bounded by (y = f(x)), the x-axis, and lines (x=a) and (x=b) is revolved about the x-axis, the volume (V) is:

( V = pi int_{a}^{b} y^2 , dx )

2. Revolution about the y-axis:

If the region bounded by (x = g(y)), the y-axis, and lines (y=c) and (y=d) is revolved about the y-axis, the volume (V) is:

( V = pi int_{c}^{d} x^2 , dy )

Example 1

Question: Find the volume of the solid formed by revolving the region bounded by (y = x), the x-axis, and the lines (x=0) and (x=2) about the x-axis.

Solution:

Step 1: Write the formula for volume of revolution about the x-axis.

( V = pi int_{a}^{b} y^2 , dx )

Step 2: Substitute the function and limits.

Since (y=x), then (y^2 = x^2). The limits are (a=0) and (b=2).

( V = pi int_{0}^{2} x^2 , dx )

Step 3: Integrate the function.

( V = pi left[ frac{x^3}{3} right]_{0}^{2} )

Step 4: Evaluate at the limits.

( V = pi left( frac{2^3}{3} – frac{0^3}{3} right) = pi left( frac{8}{3} – 0 right) = frac{8pi}{3} )

Answer: ( frac{8pi}{3} ) cubic units.

Example 2

Question: Find the volume of the solid formed by revolving the region bounded by (x = y^2), the y-axis, and the lines (y=0) and (y=1) about the y-axis.

Solution:

Step 1: Write the formula for volume of revolution about the y-axis.

( V = pi int_{c}^{d} x^2 , dy )

Step 2: Substitute the function and limits.

Since (x=y^2), then (x^2 = (y^2)^2 = y^4). The limits are (c=0) and (d=1).

( V = pi int_{0}^{1} y^4 , dy )

Step 3: Integrate the function.

( V = pi left[ frac{y^5}{5} right]_{0}^{1} )

Step 4: Evaluate at the limits.

( V = pi left( frac{1^5}{5} – frac{0^5}{5} right) = pi left( frac{1}{5} – 0 right) = frac{pi}{5} )

Answer: ( frac{pi}{5} ) cubic units.

Teaching Methods/Instructional Techniques

Discussion, Demonstration, Guided Practice, Question and Answer, Explanation, Problem Solving, Individual Practice.

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Recalling/Engaging

Teacher’s Activity: The teacher reviews basic integration by asking students to integrate simple functions. The teacher then introduces the lesson by explaining that integration has practical applications beyond finding antiderivatives, such as calculating areas and volumes.

Pupils’ Activity: Students recall basic integration rules and participate in the discussion.

Learning Point: Integration applications introduction

Step 2: Area Under a Curve

Time: 8 minutes

Teaching Skill: Explaining/Illustrating

Teacher’s Activity: The teacher explains the concept of area under a curve, showing how definite integrals are used to calculate it. The teacher uses charts to illustrate the area bounded by a curve, the x-axis, and specific lines, as well as the area between two curves. The teacher guides students through Example 1 from the Board Summary.

Pupils’ Activity: Students listen, observe the charts, ask questions, and follow the worked example.

Learning Point: Area under curve concept

Step 3: The Trapezoidal Rule

Time: 8 minutes

Teaching Skill: Explaining/Deriving

Teacher’s Activity: The teacher explains the need for numerical methods like the trapezoidal rule for approximating areas. The teacher derives the trapezoidal rule formula and explains each term, including (h) and (n).

Pupils’ Activity: Students pay attention to the explanation, understand the formula, and note the terms.

Learning Point: Trapezoidal rule formula

Step 4: Applying the Trapezoidal Rule

Time: 8 minutes

Teaching Skill: Demonstrating/Guiding

Teacher’s Activity: The teacher guides students through Example 1 of the trapezoidal rule from the Board Summary, demonstrating how to calculate (h), find the y-values, and apply the formula step-by-step. The teacher encourages students to ask questions during the process.

Pupils’ Activity: Students follow the steps, perform calculations, and ask for clarification.

Learning Point: Trapezoidal rule application

Step 5: Volume of Solids of Revolution

Time: 7 minutes

Teaching Skill: Explaining/Visualizing

Teacher’s Activity: The teacher introduces the concept of solids of revolution, explaining how a 2D curve rotated about an axis forms a 3D solid. The teacher presents the formulas for calculating the volume when revolving about the x-axis and y-axis, using diagrams if available.

Pupils’ Activity: Students listen, visualise the concept, and note the formulas for volume.

Learning Point: Solids of revolution concept

Step 6: Calculating Volume of Solids of Revolution

Time: 5 minutes

Teaching Skill: Problem-solving/Demonstrating

Teacher’s Activity: The teacher guides students through Example 1 and Example 2 for the volume of solids of revolution from the Board Summary, demonstrating the application of the formulas and the integration process.

Pupils’ Activity: Students follow the worked examples, ask questions, and practice similar calculations.

Learning Point: Volume calculation demonstration

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. State the formula for the area under a curve (y=f(x)) from (x=a) to (x=b).
2. Write down the trapezoidal rule formula.
3. State the formula for the volume of a solid formed by revolving (y=f(x)) about the x-axis.
4. Approximate the area under (y=x+1) from (x=0) to (x=2) using the trapezoidal rule with 2 strips.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Understanding area and volume

Step 8: Note-Taking

Time: 4 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides pupils/students to copy the essential Board Summary notes, including formulas and worked examples, into their notebooks.

Pupils’ Activity: Pupils/students copy the notes carefully into their notebooks.

Learning Point: Recording key concepts

Step 9: Conclusion

Time: 2 minutes

Teaching Skill: Summarising

Teacher’s Activity: The teacher summarises the main points of the lesson, reiterating the importance of integration in calculating areas and volumes, and the utility of the trapezoidal rule for approximation. The teacher encourages students to practice more problems.

Pupils’ Activity: Students listen and ask any final questions.

Learning Point: Lesson consolidation

Continuous Assessment/Further Study

Type: Homework

Instruction: Solve the following problems in your notebook:

1. Find the exact area bounded by the curve (y = x^3), the x-axis, from (x=0) to (x=2).
2. Use the trapezoidal rule with 3 strips to estimate the area under the curve (y = frac{1}{x}) from (x=1) to (x=4).
3. Calculate the volume of the solid formed by revolving the region bounded by (y = 2x), the x-axis, and the lines (x=0) and (x=3) about the x-axis.

Lesson Keywords

• Integration – A method of finding antiderivatives or areas and volumes.
• Area under a curve – The region bounded by a function’s graph, an axis, and specific lines.
• Definite integral – An integral with upper and lower limits, used to calculate exact area.
• Trapezoidal rule – A numerical method for approximating definite integrals by dividing the area into trapezoids.
• Solids of revolution – 3D shapes formed by rotating a 2D curve around an axis.
• Volume – The amount of space occupied by a 3D object.

Differentiation

For weaker learners, provide additional guided practice with simpler functions and fewer strips for the trapezoidal rule. Offer step-by-step templates for solving problems. For faster learners, introduce more complex functions or ask them to explore the error in the trapezoidal rule approximation compared to the exact integral.

Suggested Lesson Videos

For further understanding, students can search on YouTube for: “Area under curve integration SS3”, “Trapezoidal rule explanation”, “Volume of solids of revolution further maths”.

Teacher Guide for Using This Lesson Plan

Before the lesson, ensure you have prepared the charts illustrating areas and volumes, and have a calculator handy for demonstrations. Begin by briefly revising basic integration to activate prior knowledge. When teaching the trapezoidal rule, emphasize the visual interpretation of trapezoids under the curve. For solids of revolution, encourage students to visualize the 3D shape being formed. Guide students through problem-solving steps, allowing them to attempt parts of the solution. Pay close attention to common errors in applying formulas or setting limits of integration. Students should copy the Board Summary notes after the main concepts have been taught and evaluated to reinforce their learning. Provide individual support to students struggling with algebraic manipulation or formula application.