4.) Write a double integral that cepresents the surfaceR.cegionthe doublethat lies obove theto eralvatet2=7(4,4"ofUse ocomputer algeb& system the döu bleintegral,orea2.triangledrith vertices co,0), (6,0), 6,6)37t42.

Answered: 4.) Write a double integral that…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4.) Write a double integral that cepresents the surface
R.
cegion
the double
that lies obove the
to eralvate
t2=7(4,4"
of
Use o
computer algeb& system the döu ble
integral,
orea
2.
triangle
drith vertices co,0), (6,0), 6,6)
37t4
2.

Expert Solution

Step 1

Identify the limits of integration from the region.

Here, the region is a triangle which is bounded by the lines $y = x$ , $x = 0$ and $x = 6$ .

Therefore, x varies from 0 to 6 and y varies from 0 to x.

Step 2

Differentiate the function $f (x, y) = 6 x + y^{2}$ with respect to x and y, to find f_x and f_y.

$\begin{array}{rcl} f (x, y) & = & 6 x + y^{2} \\ \frac{\partial f}{\partial x} & = & 6 (1) + 0 \\ f_{x} & = & 6 \\ \frac{\partial f}{\partial y} & = & 0 + 2 y \\ f_{y} & = & 2 y \end{array}$

Thus, $f_{x} = 6$ and $f_{y} = 2 y$ .

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