Please verify my answer.
Find the average value of the function f(x) = x√(1+2x) on the interval [0,4]
Favg = 1/(b-a) ∫ from a to b of f(x) dx
= 1/(4-0) ∫ x√(1+2x) dx
= 1/4 ∫ x(1+2x)^(1/2) dx
= 1/4 [ (x^(2))/2 (2/3) (1+2x)^(3/2)]
= 1/4 [ ∫b -∫a]
= 1/4 [ 144]
= 36 (The average value)
Not quite. For
∫ x√(1+2x) dx
Let u=1+2x, so du = 2dx and x = (u-1)/2 -- then you have
∫ (u-1)/2 * √u * 1/2 du = 1/4 ∫(u^(3/2) - u^(1/2)) du
See where that takes you
Recall that ∫f(x)*g(x) ∫f(x) * ∫g(x)
also you forgot to work with the (1+2x) properly
Alright so,
= 1/(4-0) ∫ x√(1+2x) dx
u = 1+2x
du = 2 dx
1/2 du = x
= 1/4 [ (u-1)/2 (u)^(1/2) (2)] du
= 1/4 [ u^(3/2) - u^(1/2)] du
= 1/4 [ (2/5)(1+2x)^(5/2) - (2/3)(1+2x)^(3/2)]
= (1/10)(1+2x)^(5/2) - (1/6)(1+2x)^(3/2) + C
Then, [ ∫b -∫a]
198/10 + 2/30 = 298/15
not quite
du = 2 dx, so dx = 1/2 du
you're not watching things.
1/4 ∫[0,4] x√(1+2x) dx = 1/4 (∫[1,9] (u-1)/2 * u^(1/2) * 1/2 du)
= 1/16 ∫[1,9] u^(3/2) - u^(1/2) du
are you familiar with changing the limits of integration? It saves having to convert everything back to x's before evaluating.
To verify your answer, let's go through the calculation step-by-step.
First, we have the function f(x) = x√(1+2x).
To find the average value of a function on an interval [a, b], we use the formula:
Favg = 1/(b-a) * ∫ from a to b of f(x) dx.
In this case, the interval is [0,4], so a = 0 and b = 4.
Substituting the values into the formula, we have:
Favg = 1/(4-0) * ∫ from 0 to 4 of x√(1+2x) dx.
Simplifying the expression, we get:
Favg = 1/4 * ∫ from 0 to 4 of x(1+2x)^(1/2) dx.
Now, we can integrate the expression inside the integral. Using integration techniques, we find:
∫ x(1+2x)^(1/2) dx = (x^2)/2 * (2/3)(1+2x)^(3/2) + C,
where C is the constant of integration.
Now, we substitute the limits of integration, 0 and 4, into the expression:
Favg = 1/4 * [(4^2)/2 * (2/3)(1+2(4))^(3/2) - (0^2)/2 * (2/3)(1+2(0))^(3/2)].
Simplifying further:
Favg = 1/4 * [(16/2) * (2/3)(9) - (0/2) * (2/3)(1)].
Favg = 1/4 * [8 * (2/3)(9) - 0].
Favg = 1/4 * [48].
Favg = 48/4 = 12.
So, the average value of the function f(x) = x√(1+2x) on the interval [0,4] is indeed 12.
Hence, your answer of Favg = 36 is incorrect. The correct value is Favg = 12.