Find the average value g-ave of the function g on the given interval.
g(x) = 6∛x [27, 125]
To find the average value g-ave of the function g on the interval [27, 125], we need to evaluate the definite integral of g(x) over this interval and then divide it by the length of the interval.
First, let's find the definite integral of g(x) over the interval [27, 125].
∫(27 to 125) 6∛x dx
To evaluate this integral, we can use the power rule for integration:
∫x^n dx = (1/(n+1)) * x^(n+1)
Applying the power rule, we have:
∫(27 to 125) 6∛x dx = 6 * (∫(27 to 125) x^(1/3) dx)
Using the power rule, we get:
= 6 * ( (3/4) * x^(4/3) ) | (27 to 125)
Plugging in the upper and lower limits:
= 6 * ( (3/4) * 125^(4/3) - (3/4) * 27^(4/3) )
Simplifying:
= 6 * ( (3/4) * 125^(4/3) - (3/4) * 27^(4/3) )
Now, let's find the length of the interval [27, 125]:
Length = 125 - 27 = 98
Finally, to find the average value g-ave, we divide the definite integral by the length of the interval:
g-ave = [ 6 * ( (3/4) * 125^(4/3) - (3/4) * 27^(4/3) ) ] / 98
Simplify the expression where necessary using a calculator if needed.
To find the average value of a function g(x) on a given interval [a, b], you need to calculate the definite integral of g(x) over that interval and then divide it by the length of the interval (b - a).
In this case, we need to find the average value of g(x) = 6∛x on the interval [27, 125].
Step 1: Calculate the definite integral of g(x) over the interval [27, 125].
∫(27 to 125) 6∛x dx = [6/4 * (x^(4/3))]{27 to 125}
= 6/4 * (125^(4/3)) - 6/4 * (27^(4/3))
Step 2: Simplify the expression by evaluating the definite integral.
= 6/4 * (125 * (2/3)) - 6/4 * (27 * (2/3))
= (3/2) * 125^(2/3) - (3/2) * 27^(2/3)
Step 3: Calculate the average value of g(x) by dividing the definite integral by the length of the interval.
g-ave = [(3/2) * 125^(2/3) - (3/2) * 27^(2/3)] / (125 - 27)
= [(3/2) * (5 * 5)^(2/3) - (3/2) * (3 * 3)^(2/3)] / (125 - 27)
= [(3/2) * (5)^(2/3) - (3/2) * (3)^(2/3)] / (125 - 27)
≈ 5.357
Therefore, the average value g-ave of the function g(x) = 6∛x on the interval [27, 125] is approximately 5.357.
the average value over [a,b] is ∫[a..b] g(x) dx / (b-a)
= 9/2 x^(4/3) [27,125]
= 9/2 (625-81)
= ____
Find the average value g-ave of the function g on the given interval.
i think its (1224)/(49)