Final answer:
To find the volume of the solid, we need to find the area of the rectangle R first. The area of a rectangle is given by A = length * width. In this case, the length is 4 units (2 - (-1)) and the width is 2 units (1 - (-1)). So, the area of the rectangle is 4 * 2 = 8 square units. Now, we can evaluate the volume integral and find that the volume of the solid is 86 cubic units.
Step-by-step explanation:
To find the volume of the solid, we need to find the area of the rectangle R first. The area of a rectangle is given by A = length * width. In this case, the length is 4 units (2 - (-1)) and the width is 2 units (1 - (-1)). So, the area of the rectangle is 4 * 2 = 8 square units.
Now, we need to find the volume of the solid that lies under the plane and above the rectangle. We can use the formula for the volume of a solid between two surfaces, which is given by V = ∬R f(x,y) dA, where R is the region in the xy-plane bounded by the rectangle, f(x,y) is the equation of the plane, and dA is an element of area in the xy-plane. In this case, the equation of the plane is 4x + 10y - 2z + 15 = 0.
To find the volume, we integrate f(x,y) over the region R bounded by the rectangle. The integral becomes V = ∬R (15 - 4x - 10y)/2 dz dA = ∫[a,b] ∫[c,d] (15 - 4x - 10y)/2 dz dxdy, where [a,b] and [c,d] are the bounds for z and x respectively. In this case, the bounds for z are z = 15 - 4x - 10y and z = 0, and the bounds for x and y are -1 and 2, and -1 and 1 respectively.
Now, we can evaluate the volume integral V = ∫[-1,2] ∫[-1,1] ∫[0,15 - 4x - 10y] (15 - 4x - 10y)/2 dz dxdy. After calculating the triple integral, we find that the volume of the solid is 86 cubic units.