Use a triple integral to find the volume of the solid bounded by the graphs of the equations. z = 2 − y, z = 4 − y^2, x = 0, x = 3, y = 0

Annette Arroyo

Annette Arroyo

Answered question

2021-01-31

Use a triple integral to find the volume of the solid bounded by the graphs of the equations. z=2y,z=4y2,x=0,x=3,y=0

Answer & Explanation

timbalemX

timbalemX

Skilled2021-02-01Added 108 answers

Step 1
Given:
The solid bounded by the graphs of the equations:
z=2−y and z=4y2
x=0 and x=3
y=0
We have to find the volume of the solid bounded by the given graphs of the equations.
Step 2
We know that,
The limits for z is:
z=2yz=4y2
The limit for x is:
x=0 to x=3
Now we have to find the limit of y:
We have,
z=2yz=4y2
2y=4y2
y2y2=0
y22y+y2
y(y2)+1(y2)=0
(y2)(y+1)=0
(y2)=0or(y+1)=0
y=2 or y =-1
y=-1 is not possible [y=0]
Hence we get the limit for y is:
y=0 to y=2
Step 3
V=x=0x=3y=0y=2z=2yz=4y2dzdydx
=x=0x=3y=0y=2[z]2y4y2dydx
=x=0x=3y=0y=2[4y22+y]dydx
=x=0x=3y=0y=2[2y2+y]dydx
=x=0x=3[2yy33+y22]02dx
=x=0x=3[483+2]dx
=103x=0x=3dx
=103[x]03

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