Find the range of the function F(x)= integral from negative 6 to x of √(36-t^2) dt
It is b because the domain of the integral is from [-6,6], and plugging these numbers in results in a range from [0,18pi]
Karen /Alice -- please use the same name for your posts.
I will do it
I'm sure you can verify that
F(x) = x/2 √(36-x^2) - 9π
So the range is [-9,9] - 9π
my teacher said I am wrong and he gave the options to choose:
a) [0,36pi]
b) [0,18pi]
c) [-6,6]
d) [-6,0]
Which one will be? PLEASE AND HOW
To find the range of the function F(x) = ∫(from -6 to x) √(36-t^2) dt, we can follow these steps:
Step 1: Determine the integral expression:
The function F(x) represents the integral of √(36-t^2) from -6 to x. This means that for each value of x, the function calculates the area between the curve of √(36-t^2) and the x-axis, bounded by the limits of -6 and x.
Step 2: Evaluate the integral:
Integrating the expression √(36-t^2) with respect to t will give us an antiderivative function.
To integrate the square root of (36-t^2), we can make use of trigonometric substitution. Since the expression inside the square root resembles the standard form of a circle equation, we can introduce the trigonometric identity: t = 6sinθ, where θ is an angle.
By substituting t = 6sinθ, we can express the integral in terms of θ.
Step 3: Determine the limits of integration:
The original integral was from -6 to x. However, after applying the trigonometric substitution, the limits will change in terms of θ.
When t = -6, using the substitution t = 6sinθ:
-6 = 6sinθ
sinθ = -1
θ = -π/2
When t = x, using the substitution t = 6sinθ:
x = 6sinθ
So, the new limits of integration are from -π/2 to arcsin(x/6).
Step 4: Evaluate the antiderivative:
The integral expression becomes: ∫(from -π/2 to arcsin(x/6)) 6cosθ dθ.
Integrating 6cosθ gives us 6sinθ.
Step 5: Evaluate the function F(x):
Evaluating the integral from -π/2 to arcsin(x/6) will give us the antiderivative function F(x). Note that we don't need to worry about the constant of integration as we are only interested in finding the range of the function.
F(x) = 6sin(arcsin(x/6))
F(x) = 6(x/6)
F(x) = x
Step 6: Determine the range of the function:
Since F(x) = x, the range of the function F(x) is all real numbers. In other words, the function F(x) takes on any value you choose.
Therefore, the range of the function F(x) is (-∞, +∞).