To determine
To find:
The average value of the given function on the given interval.
Answer
25π
Explanation
1) Concept:
Use the formula for the average value of h on the interval [a, b].
2) Formula:
have=1b-a∫abh(x)dx
3) Given:
hx=cos4xsinx, [0, π]
4) Calculation:
Here, hx=cos4xsinx, a=0 and b=π
Substituting all the above in the formula,
have=1π-0∫0πcos4xsinxdx
Simplify.
have=1π∫0πcos4xsinxdx
Here, we need to use the substitution method.
Let u=cosx, then du=-sinxdx, so sinxdx=-du
Since this is a definite integral, the limits of the integration also change.
When x=0, u=1 and x=π, u=-1
Therefore, the integration becomes
have=1π∫0πcos4xsinxdx
=1π∫1-1u4-du
Using the property of reversing order of definite integral,
= 1π∫-11u4du
Since u4 is an even function, use the property of definite integral for even functions.
=2π∫01u4du
Integrating,
=2πu5501
Simplifying,
=25πu501
=25π15-05
=25π
Therefore,
have=25π
Conclusion:
The average value of hx=cos4xsinx on the interval [0, π] is 25π