Question

Calculate the double integral of f(x,y) over the triangle indicated in the following figure: f(x,y)=2ye^x

Answer :Calculate the double integral of f(x,y) over the triangle indicated in the following figure: f(x,y)=2ye^x
Answer :

Answer 1

To calculate the double integral of f(x,y) = 2ye^x over a triangular region, one must determine the integration limits, set up and evaluate the integral with respect to y, and then x.

Step-by-step explanation:

To calculate the double integral of f(x,y) = 2ye^x over a given triangle, one must first establish the limits of integration. These limits are dependent on the vertex coordinates of the triangle within the x-y plane. Assuming the vertices of the triangle are known, the integration can be done by integrating with respect to y first (from a function of x, y1(x), to another function of x, y2(x)), and then integrating the result with respect to x (from x1 to x2).

The steps to complete this would be:

1. Sketch the triangle and determine the limits of integration.
2. Set up the double integral with the appropriate bounds.
3. Integrate the function with respect to y, treating x as a constant.
4. Integrate the resulting function with respect to x.

Depending on the specific triangle, some simplifications might be possible, such as linear bounds for integration. This process allows us to find the area under the surface described by f(x,y) = 2ye^x over the triangular region.

Answer 2

Without specific details on the triangle, the double integral of 2ye^x over a triangular region involves setting up the integral with appropriate limits and integrating first with respect to y and then x.

Step-by-step explanation:

To calculate the double integral of f(x,y) = 2ye^x over a triangular region, we need to set up the integral bounds based on the coordinates of the triangle's vertices. Assuming the triangle vertices are given or can be deduced from the context (which was not provided in the question), you would typically integrate with respect to y first from a linear function (or constant) to another linear function, and then integrate with respect to x from the lowest x-value of the triangle to the highest.

The double integral will be in the form:

\[ \int_{x_{min}}^{x_{max}} \int_{y_{lower}(x)}^{y_{upper}(x)} 2ye^x \, dy \, dx \]

Without the specific limits of integration, we cannot proceed with an actual calculation. However, the process involves evaluating the inner integral with respect to y, and then the outer integral with respect to x.

The complete question is: Calculate the double integral of f(x, y) over the triangle indicated in the following figure: y 4 3 2 1 f(x, y) = 12yet X 1 2 3 4 5