Hello, I'd appreciate any help with the following question below:
Information:
g(x)= 4 (x+1)^(-2/3)
f(x)= ∫ g(t) dt
The Question:
What is f(26) ?
(NOTE: I don't know how to do this on a key board, so I'll just say that while I did type an Indefinite integral for "f(x)" it is actually a Definite integral with the bounds being from 0 to "x" meaning the 0 is on the bottom of the integral sign and the x is on the top of the integral sign).
so you want
f(x)= ∫ 4 (x+1)^(-2/3) dx from 0 to x
= (4/(1/3) (x+1)^(1/3) from 0 to x
= 12(x+1)^1/3
so f(26) = 12(27)^(1/3)
= 12(3) = 36
I jumped to conclusion when I subbed in the 0
(4/(1/3) (x+1)^(1/3) from 0 to x
= 12(x+1)^1/3 from 0 to x
should have given me:
12 (x+1)^(1/3) - 12(1)^(1/3)
= 12 (x+1)^(1/3) - 12
f(x) = 12 (x+1)^(1/3) - 12
f(26) = 12(3) - 12
= 24
AH! I understand, thank you Reiny!
Sure! I can help you with that.
To find f(26), we need to evaluate the definite integral of g(t) with respect to t from 0 to 26.
Given that g(x) = 4(x+1)^(-2/3), we can compute the indefinite integral of g(t) as follows:
First, we use the power rule for integration, which states that ∫ x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration.
Applying the power rule to g(t), we have:
∫ g(t) dt = ∫ 4(t+1)^(-2/3) dt.
Now, to evaluate this definite integral from 0 to 26, we can use the Fundamental Theorem of Calculus. The theorem states that if F(x) is any function whose derivative is f(x), then ∫[a to b] f(x) dx = F(b) - F(a).
In other words, to compute a definite integral, we can find an antiderivative of the integrand and then subtract the antiderivative evaluated at the lower bound from the antiderivative evaluated at the upper bound.
In this case, we already have the antiderivative of g(t), which is:
F(t) = ∫ 4(t+1)^(-2/3) dt = -3 (t+1)^(1/3) + C.
Now, we can evaluate f(26) using the Fundamental Theorem of Calculus:
f(26) = F(26) - F(0) = [-3 (26+1)^(1/3)] - [-3 (0+1)^(1/3)].
Simplifying further, we have:
f(26) = -3 (27)^(1/3) - (-3 (1)^(1/3)).
Lastly, we can evaluate the cube root and compute the final result for f(26) by performing the subtraction.
I hope this explanation helps you understand how to find f(26) using the given information!