Integrating 1 - x^2 from -1 to 1: A Detailed Guide
When dealing with calculus, particularly integration, it is essential to understand how to integrate functions over specified intervals. This guide will walk you through the process of integrating the function 1 - x^2 from -1 to 1, providing a step-by-step explanation.
Understanding the Function and Symmetry
The function we are considering is 1 - x^2. This is a polynomial function that is defined for all real numbers. When graphed, the function is a parabola that opens downwards and is symmetric about the y-axis. This symmetry will be a key feature in simplifying the integration process.
Let's express the function step by step:
1 - x^2 1 - x^2 : for all x : in : [-1, 1]
Due to the symmetry of this function about the y-axis, we can use this property to simplify the integration from -1 to 1. Instead of integrating from -1 to 1 directly, we can integrate from 0 to 1 and then multiply the result by 2.
Detailed Integration Process
The integral we want to solve is:
( I int_{-1}^{1} (1 - x^2) dx )
Using the property of symmetry, we can rewrite the integral as:
( I 2 int_{0}^{1} (1 - x^2) dx )
Now let's break down the integration:
Integrate the function from 0 to 1: ( int_{0}^{1} (1 - x^2) dx left[ x - frac{x^3}{3} right]_{0}^{1} ) Evaluating the antiderivative at the limits of integration: ( left( 1 - frac{1^3}{3} right) - left( 0 - frac{0^3}{3} right) ) Simplifying the expression: ( 1 - frac{1}{3} frac{2}{3} )Finally, applying the factor of 2 to the result:
( I 2 cdot frac{2}{3} frac{4}{3} )
Conclusion
In summary, the integral of 1 - x^2 from -1 to 1 is equal to (frac{4}{3}). This result can be obtained using the symmetry of the function and applying the rules of definite integration.
Further Exploration
To deepen your understanding of integration and its applications, consider exploring more complex integrals and functions. Experiment with different intervals and functions, and practice applying integration techniques to real-world problems.