Calc - Find the average value of : f(x)=2sinx+7cosx on the interval [0,11/6π]

Question

Find the average value of : f(x)=2sinx+7cosx

on the interval [0,11/6π]

Please assist. I appreciate your time.

Answer 1

Average value = 1 / (11π/6 - 0)∫_{from 0 to 11π/6} [2sinx + 7cosx]dx

= (6 / (11π)) [-2cosx + 7sinx]_{(from 0 to 11π/6)}

= (6 / (11π)) [(-√3 - 7/2) - (-2 + 0)]

= (6 / (11π)) (-3/2 - √3) = (-9 - 6√3) / (11π)

Answer 2

To find the average value, we find area under the curve between x=0 and x=11*pi/6. We get

 

 ∫^11*pi/6₀(2sin(x) + 7cos(x))dt

 

We then break up the integral, as shown below:

 

 2 ∫^11*pi/6₀ sin(x) dt  + 57∫^11*pi/6₀ cos(x) dt =

 

= 2[-cos(x)] |^11*pi/6₀ +  7[sin(x)] |^11*pi/6₀

 

= (4/3)t^3/2 + 5t) |⁶₂  

 

= [-2cos(11*pi/6) + 7sin(11*pi/6)] - [-2cos(0) + 7sin(0)]

 

= [-5.23] - [-2] or [-3/2] - [√3]

= -5.23 - (-2)

= -3.23

 

The average value is -3.23