Question

Compute The Following Double Integrals. Be Sure To Show All Steps. ∬R(X2y+2)DA Where R Is The Rectangle R={(X,Y)∣−1≤X≤1,1≤Y≤2}.

Answer 1

To compute the double integral over the rectangle R={(x,y)∣−1≤x≤1,1≤y≤2} of the function f(x,y) = x^2 * y + 2, we integrate the function over the given region R and find the result to be 2/3.

Step-by-step explanation:

To compute the double integral over the rectangle R={(x,y)∣−1≤x≤1,1≤y≤2} of the function f(x,y) = x^2 * y + 2, we need to integrate the function over the given region R.

Step 1: Set up the integral:

∬R(x^2 * y + 2) dA where R is the region defined by -1≤x≤1 and 1≤y≤2.

Step 2: Perform the integration:

∫-11 ∫12 (x^2 * y + 2) dy dx

= ∫-11 [(1/2)x^2 * y^2 + 2y] from 1 to 2 dx

= ∫-11 [(1/2)x^2 * (2^2) + 2(2)] - [(1/2)x^2 * (1^2) + 2(1)] dx

= ∫-11 2x^2 + 10 dx

= [x^3/3 + 10x] from -1 to 1

= [(1/3) + 10(1)] - [(-1/3) + 10(-1)]

= (1/3 + 10) - (-1/3 + 10)

= 1/3 + 10 + 1/3 - 10

= 2/3