Question

Evaluate the line integral, where c is the given curve. (x + 8y) dx + x2 dy, c c consists of line segments from (0, 0) to (8, 1) and from (8, 1) to (9, 0).

Answer 1

Parameterize the path from (0, 0) to (8, 1) by

• x(t) = 8t and y(t) = t

and the path from (8, 1) to (9, 0) by

• x(t) = 8 + t and y(t) = 1 - t

both with 0 ≤ t ≤ 1.

Over the first path, we have

`\displaystyle \int_(C_1) (x+8y) \, dx + x^2 \, dy = \int_0^1 (8t + 8\cdot t) \cdot (8 \, dt) + (8t)^2 \cdot dt = \int_0^1 (128t + 64t^2) \, dt = \frac{256}3`

and over the second path,

`\displaystyle \int_(C_2) (x+8y) \, dx + x^2 \, dy = \int_0^1 ((8 + t) + 8(1-t)) \cdot dt + (8+t)^2 \cdot (-dt) \\\\ = \int_0^1 (-48 - 23t - t^2) \, dt = -\frac{359}6`

and so the total line integral is

`\displaystyle \int_C (x+8y) \, dx + x^2 \, dy = \frac{256}3 - \frac{359}6 = \boxed{\frac{51}2}`