Question

The temperature in degrees Celsius on the surface of a metal plate is T(x, y) = 19 − 4x2 − y2 where x and y are measured in centimeters. Estimate the average temperature when x varies between 0 and 2 centimeters and y varies between 0 and 4 centimeters. °C

Answer 1

Average value of temperature will be
`8.335^(\circ)C`

Explanation:

We have given the temperature in degree Celsius
`T(x,y)=19-4x^2-y^2`

It is given that x varies between 0 to 2

So
`0\leq x\leq 2`

And y varies between 0 and 4

So
`0\leq y\leq 4`

So area between the region A = 4×2 = 8
`cm^2`

Now average temperature is given by

`Average\ value=(1)/(A)\int \int T(x,y)dA`

`Average\ value=(1)/(8)\int \int (19-4x^2-y^2)dxdy`

`=(1)/(8)\int \int (19x-4(x^3)/(3)-y^2x)dy`

As limit of x is 0 to 2

`=(1)/(8)\int  (19* 2-4* (2^3)/(3)-y^2* 2)-(19* 0-4* (0^3)/(3)-y^2* 0))dy=(1)/(8)\int(38-(32)/(3)-2y^2)dy`

=
`=(1)/(8)(38y-(32)/(3)y-(2y^3)/(3))`

As limit of y is 0 to 4

So
`Average\ value=(1)/(8)(38* 4-(16)/(3)* 4-(2* 4^3)/(3))-0=8.335^(\circ)C`