Use symmetry to evaluate the following integral.
Integral from negative 9 to 9 left parenthesis 8 plus x plus x squared plus x cubed right parenthesis dx
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Part 1
Integral from negative 9 to 9 left parenthesis 8 plus x plus x squared plus x cubed right parenthesis dxequals
  
0  ​(Type an integer or a simplified​ fraction.)


To evaluate the integral from \(-9\) to \(9\) of the function \(8 + x + x^2 + x^3\), we can use the symmetry properties of integrals. Let's analyze each term in the integrand:

1. **Constant term \(8\)**:
   \[
   \int_{-9}^{9} 8 \, dx
   \]
   This is a constant function and is symmetric about the y-axis. Its integral is simply:
   \[
   8 \cdot (9 - (-9)) = 8 \cdot 18 = 144
   \]

2. **Linear term \(x\)**:
   \[
   \int_{-9}^{9} x \, dx
   \]
   The function \(x\) is an odd function (\(f(-x) = -f(x)\)), and the integral of an odd function over a symmetric interval \([-a, a]\) is zero. Therefore:
   \[
   \int_{-9}^{9} x \, dx = 0
   \]

3. **Quadratic term \(x^2\)**:
   \[
   \int_{-9}^{9} x^2 \, dx
   \]
   The function \(x^2\) is an even function (\(f(-x) = f(x)\)), and the integral of an even function over a symmetric interval can be simplified by doubling the integral from \(0\) to \(a\):
   \[
   \int_{-9}^{9} x^2 \, dx = 2 \int_{0}^{9} x^2 \, dx = 2 \left[ \frac{x^3}{3} \right]_0^9 = 2 \cdot \frac{9^3}{3} = 2 \cdot 243 = 486
   \]

4. **Cubic term \(x^3\)**:
   \[
   \int_{-9}^{9} x^3 \, dx
   \]
   The function \(x^3\) is an odd function, and the integral of an odd function over a symmetric interval is zero. Therefore:
   \[
   \int_{-9}^{9} x^3 \, dx = 0
   \]

Now, we sum the integrals of each term:
\[
\int_{-9}^{9} (8 + x + x^2 + x^3) \, dx = 144 + 0 + 486 + 0 = 630
\]

So, the integral evaluates to:
\[
\int_{-9}^{9} (8 + x + x^2 + x^3) \, dx = 630
\]

Would you like further details or have any questions?

Here are some related questions you might find interesting:
1. What is the integral of an odd function over a symmetric interval?
2. How can you determine if a function is even or odd?
3. How do symmetry properties simplify the evaluation of integrals?
4. What is the general formula for the integral of \(x^n\) over \([-a, a]\) when \(n\) is even?
5. What is the general formula for the integral of \(x^n\) over \([-a, a]\) when \(n\) is odd?
6. How do you handle integrals with mixed even and odd terms?
7. What are some other examples of using symmetry in calculus?
8. How do you prove that the integral of an odd function over a symmetric interval is zero?

**Tip:** When evaluating integrals, always check for symmetry. It can significantly simplify your calculations!

Learn how to evaluate the integral of the function 8 + x + x^2 + x^3 from -9 to 9 using symmetry properties. This problem covers integral calculus concepts, including the evaluation of constant, odd, and even functions over symmetric intervals. Suitable for advanced high school students.

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