Find the average value gave of the function g on the given interval.g(x) = 5 cos(x), − 𝜋2, 𝜋2
Question
Find the average value of the function
g(x) = 5 cos(x) on the interval ( [−\frac{\pi}{2}, \frac{\pi}{2}] \
Solution
The average value of a function on the interval [a, b] is given by the formula:
1/(b - a) * ∫ from a to b [f(x) dx]
Here, our function g(x) = 5cos(x) and the interval is [-π/2, π/2].
So, we need to find:
1/(π - (-π/2)) * ∫ from -π/2 to π/2 [5cos(x) dx]
This simplifies to:
2/π * ∫ from -π/2 to π/2 [5cos(x) dx]
The integral of cos(x) is sin(x), so we have:
2/π * [5sin(x)] from -π/2 to π/2
Evaluating this gives:
2/π * [5sin(π/2) - 5sin(-π/2)]
Since sin(π/2) = 1 and sin(-π/2) = -1, this further simplifies to:
2/π * [51 - 5(-1)] = 2/π * [5 + 5] = 2/π * 10 = 20/π
So, the average value of the function g(x) = 5cos(x) on the interval [-π/2, π/2] is 20/π.
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