128k views
4 votes
Find Average value of the function over the given interval y=6-x^2 [-5,2]

1 Answer

5 votes

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:

  1. Identify a and b: a = -5 and b = 2.
  2. Calculate the width of the interval (b - a): 2 - (-5) = 7.
  3. Set up the integral of the function from a to b: \(\int_{-5}^{2}(6 - x^2)dx\).
  4. Calculate the integral: The integral of 6 with respect to x is 6x, and the integral of \(-x^2\) is \(-\frac{x^3}{3}\).
  5. 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.

User Krasnoff
by
8.1k points