Question

Let r be the region bounded by the x and y axes and the line x y=1. Evaluate the double integral at xda.

a) ∫∫_r x dA
b) ∫∫_r y dA
c) ∫∫_r 1/x dA
d) ∫∫_r 1/y dA

Answer 1

To evaluate the double integral at xda, determine the limits of integration and the integrand for each variable, x and y. Use the triangular region bounded by the x and y axes and the line xy = 1. Apply the appropriate limits and integrands for each option.

Step-by-step explanation:

To evaluate the double integral at xda, we need to determine the limits of integration and the integrand for each variable, x and y. The region bounded by the x and y axes and the line xy = 1 forms a triangular region.

a) ∫∫r x dA: The limits of integration for x are from y = 0 to y = 1 and for y are from x = 0 to x = 1/y. The integrand is x.

b) ∫∫r y dA: The limits of integration for x are from y = 0 to y = 1 and for y are from x = 0 to x = 1/y. The integrand is y.

c) ∫∫r 1/x dA: The limits of integration for x are from y = 0 to y = 1 and for y are from x = 0 to x = 1/y. The integrand is 1/x.

d) ∫∫r 1/y dA: The limits of integration for x are from y = 0 to y = 1 and for y are from x = 0 to x = 1/y. The integrand is 1/y.