Table of Contents

1. Introduction
2. Finding Areas Using Multiple Integrals
   • Areas Partly Above and Below the x-axis
   • Total Areas Between Two Curves
3. Finding Areas Using Absolute Value
   • Areas Partly Above and Below the x-axis
   • Total Areas Between Two Curves
4. Practice Questions
5. Glossary
6. Summary and Key Takeaways
7. Exam Strategy
8. Real-World Applications

Introduction

In calculus, definite integrals are used to find the area under a curve. However, calculating areas can become complex when the region of interest lies partly above and partly below the x-axis or between multiple curves. This guide will help you understand how to handle these situations using multiple integrals and absolute values.

Finding Areas Using Multiple Integrals

Areas Partly Above and Below the x-axis

When a region between a curve and the x-axis lies partly above and partly below the x-axis, it is essential to calculate areas separately for each part.

Formula:

$$\text{Total Area} = \int_{a}^{b} f(x) , dx + \left| \int_{b}^{c} f(x) , dx \right|$$

Example:

To find the total area of the regions enclosed by the x-axis and the curve $y = x^3 - 12x^2 + 35x$:

1. Identify the points where the curve intersects the x-axis, $x = 0, 5, 7$.
2. Split the integral at these points: $$\text{Total Area} = \int_{0}^{5} (x^3 - 12x^2 + 35x) , dx + \left| \int_{5}^{7} (x^3 - 12x^2 + 35x) , dx \right|$$

Using a calculator or manual integration: $$\text{Total Area} = \left[ \frac{x^4}{4} - 4x^3 + \frac{35x^2}{2} \right]\_{0}^{5} + \left| \left[ \frac{x^4}{4} - 4x^3 + \frac{35x^2}{2} \right]\_{5}^{7} \right|$$ $$\text{Total Area} = 101.75 \text{ units squared}$$

Total Areas Between Two Curves

When two curves intersect at more than two points, calculate the total area by integrating the difference between the curves for each segment.

Formula:

$$\text{Total Area} = \int_{a}^{b} \left[ f(x) - g(x) \right] , dx + \int_{b}^{c} \left[ g(x) - f(x) \right] , dx$$

Worked Example:

Find the total area enclosed by the curves $y = (x-2)(x-3)^2$ and $y = x^2 - 5x + 6$:

1. Find intersection points: $x = 2, 3, 4$.
2. Set up integrals for each segment: $$\int_{2}^{3} \left[ (x-2)(x-3)^2 - (x^2 - 5x + 6) \right] , dx + \int_{3}^{4} \left[ (x^2 - 5x + 6) - (x-2)(x-3)^2 \right] , dx$$

Solving manually or using a calculator: $$\text{Area} = \frac{1}{4} + \frac{1}{4} = \frac{1}{2} \text{ units squared}$$

Finding Areas Using Absolute Value

Areas Partly Above and Below the x-axis

When using a calculator, you can integrate the absolute value of the function over the desired interval.

Formula:

$$\text{Area} = \int_{a}^{b} \left| f(x) \right| , dx$$

Example:

For the curve $y = x^3 - 12x^2 + 35x$ between $x = 0$ and $x = 7$:

Using a calculator: $$\text{Area} = \int\_{0}^{7} \left| x^3 - 12x^2 + 35x \right| \, dx = 101.75 \text{ units squared}$$

Total Areas Between Two Curves

Integrate the absolute value of the difference between the two functions over the desired interval.

Formula:

$$\text{Area} = \int_{a}^{b} \left| f(x) - g(x) \right| , dx$$

Worked Example:

For curves $y = (x-2)(x-3)^2$ and $y = x^2 - 5x + 6$:

Using a calculator: $$\text{Area} = \int\_{2}^{4} \left| (x-2)(x-3)^2 - (x^2 - 5x + 6) \right| \, dx = 0.5 \text{ units squared}$$

Practice Questions

Practice Question

1. Find the area enclosed by the curve $y = x^2 - 4x + 3$ and the x-axis between $x = 0$ and $x = 4$.

Practice Question

2. Calculate the total area between the curves $y = x^2$ and $y = x + 2$ from $x = -1$ to $x = 2$.

Glossary

• Definite Integral: The integral of a function over a specified interval, giving the net area between the function and the x-axis.
• Absolute Value: The non-negative value of a number or expression, denoted $|x|$.
• Intersection Points: Points where two curves meet or cross each other.
• Top Function: The function with the higher value in a specified interval when dealing with two curves.

Summary and Key Takeaways

• When finding areas partly above and partly below the x-axis, integrate separately and sum the absolute values of negative areas.
• For areas between two curves, integrate the difference between the top and bottom functions for each segment.
• Using absolute value in integrals simplifies calculations but requires the use of a calculator.
• Always write down the integral being evaluated to ensure full credit in exams.

Exam Strategy

• Show All Steps: Even when using a calculator, write down the integral setup for full credit.
• Check Intersection Points: Ensure you correctly identify intersection points for setting integration limits.
• Use Absolute Values: Simplify complex areas by integrating absolute values when appropriate.

Real-World Applications

• Physics: Calculating the work done by a variable force.
• Economics: Finding consumer and producer surplus.
• Engineering: Determining the area under stress-strain curves.

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By understanding and practicing these techniques, you will be well-prepared to handle complex area calculations in calculus. Happy studying!