Math Snap

STEP 1

1. The integral is a line integral over a piecewise linear path.
2. The path $L_2$ consists of two segments: from $P(1,1)$ to $R(2,1)$, and from $R(2,1)$ to $Q(2,2)$.
3. The line integral will be evaluated separately over each segment and then summed.

STEP 2

1. Parameterize the path from $P(1,1)$ to $R(2,1)$.
2. Evaluate the line integral over the first segment.
3. Parameterize the path from $R(2,1)$ to $Q(2,2)$.
4. Evaluate the line integral over the second segment.
5. Sum the results from both segments.

STEP 3

Parameterize the path from $P(1,1)$ to $R(2,1)$. This path is a horizontal line where $y = 1$ and $x$ varies from 1 to 2. We can parameterize it as:
$$x = t, \quad y = 1, \quad \text{where } t \in [1, 2]$$

STEP 4

Substitute the parameterization into the integral and evaluate over the first segment:
$$\int_{1}^{2} ((t + 1) \, dt + (1 - t) \cdot 0)$$ $$= \int_{1}^{2} (t + 1) \, dt$$ Calculate the integral:
$$= \left[ \frac{t^2}{2} + t \right]_{1}^{2}$$ $$= \left( \frac{2^2}{2} + 2 \right) - \left( \frac{1^2}{2} + 1 \right)$$ $$= (2 + 2) - \left( \frac{1}{2} + 1 \right)$$ $$= 4 - \frac{3}{2}$$ $$= \frac{8}{2} - \frac{3}{2}$$ $$= \frac{5}{2}$$

STEP 5

Parameterize the path from $R(2,1)$ to $Q(2,2)$. This path is a vertical line where $x = 2$ and $y$ varies from 1 to 2. We can parameterize it as:
$$x = 2, \quad y = t, \quad \text{where } t \in [1, 2]$$

STEP 6

Substitute the parameterization into the integral and evaluate over the second segment:
$$\int_{1}^{2} ((2 + t) \cdot 0 + (t - 2) \, dt)$$ $$= \int_{1}^{2} (t - 2) \, dt$$ Calculate the integral:
$$= \left[ \frac{t^2}{2} - 2t \right]_{1}^{2}$$ $$= \left( \frac{2^2}{2} - 2 \times 2 \right) - \left( \frac{1^2}{2} - 2 \times 1 \right)$$ $$= (2 - 4) - \left( \frac{1}{2} - 2 \right)$$ $$= -2 - \left( \frac{1}{2} - 2 \right)$$ $$= -2 + \frac{3}{2}$$ $$= -\frac{4}{2} + \frac{3}{2}$$ $$= -\frac{1}{2}$$

Sum the results from both segments to find the total integral:
$$I_2 = \frac{5}{2} + \left(-\frac{1}{2}\right)$$ $$= \frac{5}{2} - \frac{1}{2}$$ $$= \frac{4}{2}$$ $$= 2$$ The value of the integral $I_2$ is:
$$\boxed{2}$$