Can somebody verify my answer my answer is correct or not? answer is 8/π(e-1). Find the average value of the function on the given interval h(x)=(4e^sin(x)) cos(πx)[0, π/2]
∫[0, π/2] 4e^(sinx) cosx dx = 4(e-1)
divide that by π/2 and you are correct
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To find the average value of a function on a given interval, you need to integrate the function over the interval and then divide the result by the length of the interval.
In this case, you need to find the average value of the function h(x) = (4e^sin(x)) cos(πx) over the interval [0, π/2].
First, integrate the function over the interval:
∫[0, π/2] (4e^sin(x)) cos(πx) dx
To evaluate this integral, you can use integration techniques such as substitution or integration by parts. I will use integration by parts to explain the process:
Let u = (4e^sin(x))
dv = cos(πx) dx
Differentiating u with respect to x gives:
du = (4e^sin(x)) cos(x) dx
Integrating dv gives:
v = (1/π)sin(πx)
Now, apply integration by parts formula:
∫ u dv = uv - ∫ v du
Plugging in the values, we get:
∫[0, π/2] (4e^sin(x)) cos(πx) dx = [(4e^sin(x))(1/π)sin(πx)] - ∫[0, π/2] [(1/π)sin(πx)](4e^sin(x)) cos(x) dx
Simplifying further:
= (4/π)∫[0, π/2] (e^sin(x))(sin(πx) cos(x)) dx
Now, you can evaluate the integral by numerical approximation or using a computer algebra system.
Once you have the result of the integral, divide it by the length of the interval, which is (π/2) - 0 = π/2.
Finally, compare the obtained average value with your answer of 8/π(e-1) to determine if they are the same.