Evaluate the integral from -2 to 2 of (x^3 cos(x/2) + 1/2)√(4 - x^2) dx. The Wi-Fi password is the first 10 digits of the answer.
Understand the Problem
The question involves evaluating a definite integral with a specific range and is asking for the first 10 digits of the result to use as a Wi-Fi password.
Answer
3.1415926535
Answer for screen readers
The first 10 digits of the result are given by:
$$ \pi \approx 3.1415926535 $$
Steps to Solve
- Identify the integral and the function The integral to evaluate is
$$ I = \int_{-2}^{2} \left( x^3 \cos\left(\frac{x}{2}\right) + \frac{1}{2} \right) \sqrt{4 - x^2} , dx $$
- Analyze the symmetry of the function Recognize that the integrand consists of two parts: $x^3 \cos\left(\frac{x}{2}\right)$ and $\frac{1}{2}$.
- The term $x^3 \cos\left(\frac{x}{2}\right)$ is an odd function because $x^3$ is odd and $\cos\left(\frac{x}{2}\right)$ is even.
- The term $\frac{1}{2}$ is a constant (even function).
Since the integral of an odd function over a symmetric interval around zero is zero:
$$ \int_{-2}^{2} x^3 \cos\left(\frac{x}{2}\right) \sqrt{4 - x^2} , dx = 0 $$
- Evaluate the remaining integral Now we only need to evaluate the integral of the remaining part:
$$ I = \int_{-2}^{2} \frac{1}{2} \sqrt{4 - x^2} , dx $$
- Simplify the integral Factor out the constant $\frac{1}{2}$:
$$ I = \frac{1}{2} \int_{-2}^{2} \sqrt{4 - x^2} , dx $$
- Recognize the integral as the area of a semicircle The function $\sqrt{4 - x^2}$ represents the upper half of a circle with radius 2 centered at the origin. The area of a semicircle is given by:
$$ \text{Area} = \frac{1}{2} \pi r^2 $$
Here, $r = 2$, so:
$$ \text{Area} = \frac{1}{2} \pi (2)^2 = \frac{1}{2} \cdot \pi \cdot 4 = 2\pi $$
- Finish computing the integral Now substitute back into our expression for $I$:
$$ I = \frac{1}{2} (2\pi) = \pi $$
The first 10 digits of the result are given by:
$$ \pi \approx 3.1415926535 $$
More Information
The result of the integral corresponds to the area of a semicircle. This mathematical elegance is associated with many real-world applications, particularly in physics and engineering.
Tips
- Misidentifying the symmetry of the function and failing to recognize that the contribution from the odd function part of the integral is zero.
- Forgetting to factor out constants when evaluating definite integrals.