CMIIh
2021-02-25
Answered

Use Green's Theorem to evaluate ${\int}_{C}({e}^{x}+{y}^{2})dx+({e}^{y}+{x}^{2})dy$ where C is the boundary of the region(traversed counterclockwise) in the first quadrant bounded by $y={x}^{2}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}y=4$ .

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Szeteib

Answered 2021-02-26
Author has **102** answers

Step 1

The give integral is

where C is the boundary of the region in first quadrant bounded by

From greens theorem we know that

Comparing we get

So,

Step 2

From (1), (2) and (3) we get

Now, we will find the region of integration.

this gives

Since the region lies on first quadrant so we leave negative sign.

Therefore, x varies from 0 to 2 and y varies from 4 to x2

Now, from (4)

Step 4

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S is the surface of the solid bounded by the cylinder