Question

Determine the value of the double integral 1. I=11 I=∬ R (4−x)dxdy 2. I=13 over the region 3. I=14 R={(x,y):2≤x≤4,0≤y≤6} 4. I=12 in the xy-plane by first identifying it as the 5. I=10 volume of a solid.

Answer 1

To determine the value of the double integral of (4-x) over the region R, which is bounded by 2≤x≤4 and 0≤y≤6, we perform sequential integration first in the x-direction and then in the y-direction. The result is the volume of the corresponding solid.

Step-by-step explanation:

We need to determine the value of the double integral of the function (4-x) over the region R defined by 2≤x≤4 and 0≤y≤6 in the xy-plane. This region R corresponds to a rectangular area in the xy-plane. To find the volume of the solid represented by this integral, we will integrate the given function across the specified limits.

The double integral can be set up as follows:

1. Integrate (4-x) with respect to x from 2 to 4.
2. Integrate the result with respect to y from 0 to 6.

The integration will be performed in two steps:

1. ∫ f(x) dx from x=2 to x=4, where f(x) = (4-x).
2. ∫ [Result from step 1] dy from y=0 to y=6.

The first integral is with respect to x and we integrate (4-x) from 2 to 4. This gives us a result which we can then integrate with respect to y from 0 to 6 to get the final volume of the solid.