determine the average value of f(x) over the interval from x=a to x=b, where f(x)=100e^-.5x; a=0, b=4
To determine the average value of f(x) over the interval from x=a to x=b, we need to calculate the definite integral of f(x) over that interval and then divide it by the length of the interval.
First, let's calculate the definite integral of f(x) from x=a to x=b:
∫[a to b] f(x) dx = ∫[0 to 4] 100e^(-0.5x) dx
To evaluate this integral, we can use the substitution method. Let's make the substitution u = -0.5x, which implies du = -0.5 dx:
∫ 100e^(-0.5x) dx = ∫ 100e^u (-2 du)
= -200 ∫ e^u du
= -200e^u + C
Now, let's evaluate this integral from x=0 to x=4:
∫[0 to 4] 100e^(-0.5x) dx = [-200e^u] from x=0 to x=4
= -200e^(-0.5*4) - (-200e^(-0.5*0))
= -200e^-2 + 200e^0
= -200e^-2 + 200
Next, let's calculate the length of the interval, which is b - a:
Length of interval = b - a
= 4 - 0
= 4
Finally, let's compute the average value of f(x) over the interval:
Average value = (Definite integral of f(x) over the interval) / (Length of interval)
= (-200e^-2 + 200) / 4
Therefore, the average value of f(x) over the interval from x=0 to x=4 is (-200e^-2 + 200) / 4.
To determine the average value of a function over an interval, you need to calculate the definite integral of the function over that interval and then divide the result by the width of the interval.
In this case, the function is f(x) = 100e^(-0.5x) and the interval is from a = 0 to b = 4.
Step 1: Calculate the definite integral of f(x) over the interval [a, b]. To do this, integrate the function with respect to x and evaluate it from a to b:
∫[a,b] f(x) dx = ∫[0,4] 100e^(-0.5x) dx
Step 2: Integrate the function:
= 100 * ∫[0,4] e^(-0.5x) dx
Step 3: Use integration rules to solve the definite integral. The integral of e^(-0.5x) can be found using the rule ∫e^ax dx = (1/a) * e^ax:
= 100 * (1/-0.5) * e^(-0.5x) | from 0 to 4
Step 4: Evaluate the integral at the upper and lower limits:
= 100 * (-2) * (e^(-0.5*4) - e^(-0.5*0))
Step 5: Simplify the expression:
= -200 * (e^(-2) - 1)
Step 6: Calculate the width of the interval, which is b - a:
Width = b - a = 4 - 0 = 4
Step 7: Calculate the average value of f(x) over the interval by dividing the integral by the width:
Average Value = [-200 * (e^(-2) - 1)] / 4
Step 8: Simplify the expression to find the average value:
Average Value = -50 * (e^(-2) - 1)
Therefore, the average value of f(x) over the interval from x = 0 to x = 4, where f(x) = 100e^(-0.5x), is approximately -50 * (e^(-2) - 1).
(1/4) integral from 0 to 4 of 100 e^-.5 x dx
25 int e^-.5x dx
25 (-2)e^-.5x at 4 - at 0
-50 [ e^-2 - e^0]
-50 [ .135 - 1 ]
-50 [ -.864 ]
43.2