Find the average value of sqrt(cosx)on the interval [-1,1]
Integrate it, then divide by 2. I will be happy to critique your thinking.
Here is what I get when I integrate it:[(2/3)(cosx)^(3/2)(-sinx)]
Now I plug the interval into it:
(1/2)[[(2/3)(cos1)^(3/2)(-sin1)]-[(2/3)(cos(-1))^(3/2)(-sin(-1))]]
I did something wrong, since the answer is suppose to be .914
If you differentiate what you got for the integral, you should get sqrt(cosx). I didn't get that. You may want to do some numerical integration, your calculator should be able to do that. Or you can do it with Simpson's method.
I haven't learned about the Simpson's method.
What type of calculator do u currently have?
To find the average value of sqrt(cosx) on the interval [-1,1], you need to compute the definite integral of sqrt(cosx) over that interval and then divide the result by the length of the interval.
Step 1: Compute the definite integral
To find the integral, we can use any standard integration technique. In this case, we'll make use of a substitution. Let u = cos(x), then du = -sin(x)dx. Rearranging the equation gives us dx = -du/sin(x).
The integral can now be expressed as:
∫ (sqrt(cosx))dx = ∫ (sqrt(u) * (-du)/sin(x)) = -∫ (sqrt(u)/sin(u)) du.
Step 2: Evaluate the integral
The integral ∫ (sqrt(u)/sin(u)) du is a non-elementary integral, which means it can't be solved using basic algebraic functions. In this case, we can use numerical methods or a computer program to approximate the value of the integral.
Step 3: Divide by the length of the interval
The length of the interval [-1,1] is given by 1 - (-1) = 2. Divide the value from Step 2 by 2 to get the average value of sqrt(cosx) on the interval.
It's worth mentioning that using numerical methods or computer programs, such as numerical integration or software like MATLAB or Wolfram Alpha, can help calculate the definite integral and find the average value quickly and accurately.