Evaluate the line integral by the two following methods. xy dx + x2y3 dy c is counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 2) (a) directly (b) using green's theorem

The line integral in both cases is  

a) The line integral is     where C is counterclockwise around the triangle with vertices  and  .

The boundaries of integration is shown in the attached image.

Our first line integral is

The equation of this line is  ,   varies from to .

Our second line integral is

The equation of this line is  and varies from to .

The third line integral is

The equation of this line is  and varies from to .

The value of the line integral is

b) The second part requires the use of Green's Theorem to evaluate:

This is given by:

We choose our region of integration parallel to the -axis.

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a)

b)

Step-by-step explanation:

a) The first part requires that we use line integral to evaluate directly.

The line integral is

where C is counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 2)

The boundary of integration is shown in the attachment.

Our first line integral is

The equation of this line is y=0, x varies from 0 to 1.

When we substitute y=0 every becomes zero.

Our second line integral is

The equation of this line is:

y varies from 1 to 2.

We substitute the boundary and the values to get:

The 3rd line integral is:

The equation of this line is

x varies from 0 to 1.

We substitute to get:

The value of the line integral is

b) The second part requires the use of Green's Theorem to evaluate:

Since C is a closed curve with counterclockwise orientation, we can apply the Green's Theorem.

This is given by:

We choose our region of integration parallel to the y-axis.