Question

integrate f(u,v) = v - sqrt(u) over the triangular region cut from the first quadrant on the uv plane by the line u+v=49

Answer 1

The value of the integral of
`\(f(u,v) = v - √(u)\)` over the triangular region cut from the first quadrant on the uv plane by the line
`\(u+v=49\) is \(784/3\).`

Step-by-step explanation:

To solve this problem, we'll start by setting up the integral bounds. The line
`\(u+v=49\)` intersects the u and v axes at the points
`\((49, 0)\)` and
`\((0, 49)\)`respectively, forming a triangle in the first quadrant. Integrating
`\(f(u,v) = v - √(u)\)` over this triangular region involves breaking the integral into two parts: integrating with respect to v first and then u.

For the inner integral with respect to v, the bounds are from 0 to
`\(49-u\)` as v varies from the line
`\(u+v=49\)` to the v-axis. The outer integral with respect to u ranges from 0 to 49, which are the limits of u within the triangular region.

Now, performing the integration steps:

`\(\int_(0)^(49)\int_(0)^(49-u) (v - √(u)) \,dv\,du\)`

First, integrating with respect to \(v\) yields:

`\(\left[(v^2)/(2) - v√(u)\right]_(0)^(49-u) = \left(((49-u)^2)/(2) - (49-u)√(u)\right)\)`

Then integrating this expression with respect to
`\(u\) from \(0\) to \(49\):`

`\(\int_(0)^(49)\left(((49-u)^2)/(2) - (49-u)√(u)\right) \,du = (784)/(3)\)`

Therefore, the final answer to the integral over the triangular region is
`\(784/3\).`

Answer 2

To integrate the given function over the triangular region, we can use a change of variables and then perform the integration term by term.

Step-by-step explanation:

To evaluate the integral of f(u,v) = v - sqrt(u) over the triangular region cut from the first quadrant by the line u+v=49, we can use a change of variables. Let's substitute u = x and v= 49 - x, where x represents the new variable.

The integral becomes ∫[(49 - x) - sqrt(x)] dx over the interval from 0 to 49. Simplifying this expression further, we get ∫(49 - 2x - sqrt(x)) dx. We can now integrate term by term to find the solution.