Area Between a Curve and the x-Axis

Table of Contents

1. Introduction
2. Finding the Area Between a Curve and the x-Axis
3. Handling Unknown Limits
4. Negative Areas
5. Exam Tips
6. Practice Questions
7. Glossary
8. Summary and Key Takeaways

Introduction

In this section, we will explore how to find the area between a curve and the x-axis using definite integrals. We will cover the process for both known and unknown limits and discuss what happens when the area is negative.

Finding the Area Between a Curve and the x-Axis

To find the area between a curve $y=f(x)$ and the x-axis from $x=a$ to $x=b$, you need to calculate the definite integral of the function between these limits.

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

Note: This method works as long as $f(x) \ge 0$ on the interval $[a, b]$. The result will give you the area between the curve and the x-axis from $x=a$ to $x=b$.

Example

Consider the function $y = 5 + 2x - x^2$. To find the area between this curve and the x-axis from $x=1$ to $x=3$:

$$\int_{1}^{3} (5 + 2x - x^2) , dx = \frac{28}{3} , \text{square units}$$

Handling Unknown Limits

If the limits are not provided, they are often the $x$-axis intercepts of the function. To find the $x$-axis intercepts, set $y = 0$ and solve for $x$.

Example

For the function $y = x(5 - x)$, the $x$-axis intercepts are found by setting:

$$x(5 - x) = 0 \implies x = 0 \text{ or } x = 5$$

To find the area under this curve between $x=0$ and $x=5$:

$$\int_{0}^{5} x(5 - x) , dx = \frac{125}{6} , \text{square units}$$

Note: The $y$-axis (i.e., $x=0$) may also be one of the limits.

Negative Areas

If the area lies below the x-axis, the value of the definite integral will be negative. However, the area itself is always positive.

Example

For the function $y = x^2 - 6x + 5$ between $x=2$ and $x=4$:

$$\int_{2}^{4} (x^2 - 6x + 5) , dx = -\frac{22}{3}$$

The area is the absolute value of the definite integral:

$$\text{Area} = \left| -\frac{22}{3} \right| = \frac{22}{3} , \text{square units}$$

Practice Questions

Practice Question

1. Find the area between the curve $y = x^3 - 4x$ and the x-axis from $x=-2$ to $x=2$.

Practice Question

2. Calculate the area under the curve $y = \sin(x)$ from $x=0$ to $x=\pi$.

Practice Question

3. Determine the area between the curve $y = e^x$ and the x-axis from $x=0$ to $x=1$.

Glossary

• Definite Integral: The integral of a function over a specific interval, providing the net area between the curve and the x-axis.
• Accumulation of Change: The concept of summing small changes over an interval to find the total change.
• Modulus (Absolute Value): The non-negative value of a number, disregarding its sign.

Summary and Key Takeaways

• The area between a curve $y=f(x)$ and the x-axis from $x=a$ to $x=b$ is found using the definite integral $\int_{a}^{b} f(x) , dx$.
• If the limits are unknown, solve for the $x$-axis intercepts.
• Areas below the x-axis result in negative definite integrals; take the absolute value to find the actual area.
• Always differentiate between finding the value of an integral and the area under a curve.

Key Takeaways

1. Use definite integrals to calculate areas between curves and the x-axis.
2. Ensure you take the absolute value of the integral for areas below the x-axis.
3. Understand the difference between integral values and areas.

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