Calculate the iterated integral. \int_{0}^{4}\int_{0}^{12}2e^{x+3y}dx\ dy

melodykap 2021-01-04 Answered
Calculate the iterated integral.
040122ex+3ydx dy
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Expert Answer

SabadisO
Answered 2021-01-05 Author has 108 answers
Step 1
I=040122ex+3ydx dy
Consider the inner integral, name it as I1
I1=0122e(x+3y)dx
Take the constant 2 out of the integral sign.
I1=2012e(x+3y)dx
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=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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