What is the average value of the function # f(x)=e^(-x)*sin(x)# on the interval #[1, pi]#?

Answer 1

I get #1/(pi-1) int_1^pi e^-x * sinx dx = (e+e^pi(sin(1)+cos(1)))/(2(pi-1)e^(pi+1))#

Using the definition of a function's average value, we must determine

#1/(pi-1) int_1^pi e^-x * sinx dx#.
Evaluate #int e^-x * sinx dx# by parts.
Use #u = sinx# and #dv = e^-x dx# (Or the other way around. In this case, either will work.)
We get #du = cosx dx# and #v = -e^-x#.
#uv-int v du = -e^-xsinx + int e^-xcosxdx#
Repeat parts with #u = cosx# and #dv = e^-x dx# for this integral to get
#int e^-x * sinx dx = -e^-xsinx - e^-x-inte^-xsinx dx#.
Add the integral to both sides and divide by #2# then factor, to finish with
#int e^-x sinx dx = -1/(2e^x)(sinx+cosx)#.
Evaluating from #1# to #pi#, we get:
#int_1^pi e^-x sinx dx = -1/(2e^x)(sinx+cosx)]_1^pi#
# = 1/(2e^pi) + (sin(1)+cos(1))/(2e)#
# = (e+e^pi (sin(1)+cos(1)))/(2e^(pi+1))#.

Lastly, divide the result by the interval's length to find the answer:

#1/(pi-1) int_1^pi e^-x sinx dx = (e+e^pi(sin(1)+cos(1)))/(2(pi-1)e^(pi+1))#.
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Answer 2

To find the average value of the function (f(x) = e^{-x} \cdot \sin(x)) on the interval ([1, \pi]), you can use the formula for the average value of a function over an interval:

[ \text{Average value} = \frac{1}{b - a} \int_{a}^{b} f(x) , dx ]

where (a) and (b) are the endpoints of the interval.

In this case, (a = 1) and (b = \pi). So, we can plug these values into the formula:

[ \text{Average value} = \frac{1}{\pi - 1} \int_{1}^{\pi} e^{-x} \cdot \sin(x) , dx ]

Then, you would need to evaluate the integral to find the average value.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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