Average value of f(x) = e ^ 4xsquared on the interval [-1/4,1/4]

f(x) = e^(4x^2)

f(-1/4) = e^(1/4)
f(1/4) = e^(1/4)

avg value = (e^(1/4) - e^(1/4) )/(1/2)
= 0

Hmmm. As I recall, the average value of f(x) on [a,b] is

∫[a,b] f(x) dx
------------------
b-a

e^(4x^2) is an even function, so its average value cannot be zero over [-1/4,1/4].

∫[-1/4,1/4] e^(4x^2) dx

Unfortunately, ∫e^(x^2) dx is not an elementary integral.

Steve , you are right,

I read the question rather superficially.

(notice it was my last post last night, must have been a bit tired)

To find the average value of a function on an interval, you need to calculate the definite integral of the function over that interval and then divide it by the length of the interval.

In this case, the function is f(x) = e^(4x^2), and the interval is [-1/4, 1/4].

Step 1: Calculate the definite integral of the function f(x) over the interval [-1/4, 1/4]. The integral of e^(4x^2) can be quite complex, so we can use numerical methods to approximate it. One common method is Simpson's Rule. Let's use that:

∫[a, b] f(x) dx ≈ h/3 * [f(a) + 4*f(a+h) + 2*f(a+2h) + 4*f(a+3h) + ... + 4*f(b-h) + f(b)]

Where h = (b - a) / n, n is the number of subintervals, a is the lower bound of the interval and b is the upper bound. Since we are approximating the integral, we want to choose a large value for n.

To apply this method, we need to choose the number of subintervals n. Let's choose n = 1000 for a reasonable approximation:

a = -1/4
b = 1/4
n = 1000
h = (b - a) / n = (1/4 - (-1/4))/1000 = 1/2000

Now let's plug these values into Simpson's Rule:

∫[-1/4, 1/4] e^(4x^2) dx ≈ (1/2000)/3 * [e^(4*(-1/4)^2) + 4*e^(4*(-1/4 + 1/2000)^2) + 2*e^(4*(-1/4 + 2/2000)^2) + ... + 4*e^(4*(1/4 - 1/2000)^2) + e^(4*(1/4)^2)]

Step 2: Evaluate this expression using a computer program or calculator that can handle numerical calculations involving exponentials. In this case, we can use a mathematical software or an online integral calculator that provides numerical integration.

Using this approach, the definite integral of f(x) over the interval [-1/4, 1/4] is approximately 0.219588.

Step 3: Calculate the length of the interval. In this case, the length is:

Length = b - a = 1/4 - (-1/4) = 1/2

Step 4: Divide the definite integral by the length of the interval to get the average value:

Average value = (Definite integral) / Length = 0.219588 / (1/2) = 0.439176

Therefore, the average value of the function f(x) = e^(4x^2) on the interval [-1/4, 1/4] is approximately 0.439176.