Question

Let F→=8xe^yi +4x2e^yj G→=8(x−y)i→ + 4(x + y)j→. let C be the path consisting of lines from (0,0) to (7,0) to (7,3) to (0,0). find each of the following integrals exactly:

(a) integral_C F dr = ______
(b) integral_C F dr = ______

Answer 1

(a) 288
`e^3` - 224

(b) -288
`e^3` + 224

In the given vector field F→ =
`8xe^yi + 4x^2e^yj` and path C defined by points (0,0) to (7,0) to (7,3) to (0,0), we can evaluate the line integrals using the parameterization of each line segment.

Step-by-step explanation:

The line integral of a vector field along a path is calculated by parameterizing the path and integrating the dot product of the vector field and the tangent vector to the path. In this case, the vector field is
`\( \mathbf{F} = 8xe^(y)\mathbf{i} + 4x^2e^(y)\mathbf{j} \)` and the path ( C ) consists of three line segments: (1) from (0,0) to (7,0), (2) from (7,0) to (7,3), and (3) from (7,3) to (0,0).

(a) For the first line segment from (0,0) to (7,0), parameterize it as
`\( \mathbf{r}(t) = \langle t, 0 \rangle \)`where
`\( 0 \leq t \leq 7 \)`. The line integral becomes

`\[ \int_C \mathbf{F} \cdot d\mathbf{r} = \int_0^7 (8te^0 + 4t^2e^0)dt = \int_0^7 (8t + 4t^2)dt = 288e^3 - 224 \]`

(b) For the second line segment from (7,0) to (7,3), parameterize it as
`\( \mathbf{r}(t) = \langle 7, t \rangle \)` where
`\( 0 \leq t \leq 3 \)`. The line integral becomes

`\[ \int_C \mathbf{F} \cdot d\mathbf{r} = \int_0^3 (8 \cdot 7e^t + 4 \cdot 7^2e^t)dt = -288e^3 + 224 \]`

The overall line integral is the sum of these contributions along each segment.

Answer 2

To find the integrals, we need to evaluate integral_C F dr and integral_C G dr. We can split the path into three segments and parametrize each segment to calculate the integrals.

Step-by-step explanation:

To find the integrals, we need to evaluate integral_C F dr and integral_C G dr. Since C consists of different lines, we can split the path into three segments: from (0,0) to (7,0), from (7,0) to (7,3), and from (7,3) to (0,0).

For the first segment, from (0,0) to (7,0), we can parametrize the path as r(t) = (t, 0) where t goes from 0 to 7.

For the second segment, from (7,0) to (7,3), we can parametrize the path as r(t) = (7, t) where t goes from 0 to 3.

For the third segment, from (7,3) to (0,0), we can parametrize the path as r(t) = (7 - t, 3 - t) where t goes from 0 to 7.