Calculate the iterated integral. int_0^4int_0^12 2e^(x+3y)dxdy

Haven 2021-01-15 Answered
Calculate the iterated integral.
040122ex+3ydxdy
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Expert Answer

Benedict
Answered 2021-01-16 Author has 108 answers
I=040122e(x+3y)dxdy
Consider the inner integral, name it as I1
I1=0122ex+3ydx
Take the constant 2 out of the integral sign.
I1=2012ex+3ydx
Step 2
Use substitution,
Substitute t = x + 3y
dt=dx
Change the limits as per the new variable,
When x0,t3y
When x12,t12+3y
Rewrite and solve the new integral as shown,
I1=23y12+3yetdt
=2(et]3y12+3y
=2(e12+3ye3y)
Step 3
Use the value of I1 and solve the outer integral.
I=042(e12+3ye3y)dy
Separate the two integrals.
I=204e12+3ydy204e3ydy
I=2e1204e3ydy204e3ydy
Evaluate the integrals,
I=23e12(e3y]0423(e3y]04
I=23e12(e12e0)23(e12e0)
I=23e12(e121)23(e121)
Step 4
Combine the terms,

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