find double integral (x+lny)dydx R= [1,3]x[7,10]

Question

find double integral (x+lny)dydx R= [1,3]x[7,10]

I keep getting the answer 6 +20ln(10) + 28ln(7)

Answer 1

∫∫_R (x+ln(y))dA = ∫₁³∫₇¹⁰ (x+ln(y))dydx

First, let's determine ∫₇¹⁰ (x+ln(y))dy:

∫₇¹⁰ (x+ln(y))dy

= xy+yln(y)-y [7,10]

= (10x+10ln(10)-10) - (7x+7ln(7)-7)

= 3x+10ln(10)-7ln(7)-3

Next, we determine ∫₁³ 3x+10ln(10)-7ln(7)-3 dx:

∫₁³ 3x+10ln(10)-7ln(7)-3 dx

= (3/2)x²+10ln(10)x-7ln(7)x-3x [1,3]

= (3/2)(3)²+10ln(10)(3)-7ln(7)(3)-3(3) - (3/2)(1)²+10ln(10)(1)-7ln(7)(1)-3(1)

= (27/2 + 30ln(10) - 21ln(7) - 9) - (3/2 + 10ln(10) - 7ln(7) - 3)

= 12 + 20ln(10) - 14ln(7) - 6

= 6 + 20ln(10) - 14ln(7)

It looks like your mistake was only in the last term. I'm guessing you forgot to change signs when getting your result.

Hope this helped!