#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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