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Find the volume of the solid bounded above by the given function over the specified region

f(x,y)=102x+y, with Ω the region from Exercise 21

Short Answer

Expert verified

The volume is :

10e+e24-494cubic units.

Step by step solution

01

Step 1. Given information

The function:

f(x,y)=10-2x+y

The region in execise 21

02

Step 2. To setup integral to find volume

In the given region we can see that

0yex0x1

So volume of the function over this region is:

V=010exf(x,y)dydx=010ex10-2x+ydydx

03

Step 3. To evaluate integral.

First integrate the function with respect to y.

0ex10-2x+ydy=10y-2xy+y220ex=10ex-2xex+e2x2-0=10ex-2xex+e2x2

Now integrate with respect to x.

0110ex-2xex+e2x2dx=10ex-2xex-ex+e2x401=10e-2(e-e)+e24-10-2(0-1)+14=10e-0+e24-10-2-14=10e+e24-494

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