Final answer:
The average value of the function y = 6 - x^2 over the interval [-5,2] is found by integrating the function over the interval and dividing by the interval's width. The calculated average value is approximately 11.57.
Step-by-step explanation:
The question asks to find the average value of the function y = 6 - x^2 over the interval [-5,2]. The average value of a continuous function y = f(x) over the interval [a, b] is given by the formula:
\[ \text{Average value} = \frac{1}{b-a} \int_{a}^{b} f(x)\, dx \]
Applying this formula to the given function and interval, we have:
- Identify a and b: a = -5 and b = 2.
- Calculate the width of the interval (b - a): 2 - (-5) = 7.
- Set up the integral of the function from a to b: \(\int_{-5}^{2}(6 - x^2)dx\).
- Calculate the integral: The integral of 6 with respect to x is 6x, and the integral of \(-x^2\) is \(-\frac{x^3}{3}\).
- Evaluate the integral from -5 to 2:
\[ \int_{-5}^{2}(6 - x^2)\, dx = [6x - \frac{x^3}{3}]_{-5}^{2}
\[ = (6(2) - \frac{2^3}{3}) - (6(-5) - \frac{(-5)^3}{3}) \]
\[ = (12 - \frac{8}{3}) - (-30 - \frac{-5^3}{3}) \]
\[ = 12 - \frac{8}{3} + 30 + \frac{125}{3} \]
\[ = 42 + \frac{117}{3} \]
\[ = 42 + 39 \]
\[ = 81 \]
Divide the integral by the width of the interval (b - a):
\[ \text{Average value} = \frac{81}{7} \]
\[ = 11.5714 \]
Therefore, the average value of the function y = 6 - x^2 over the interval [-5,2] is approximately 11.57.