Solving for the Volume of a Solid Using Double Integrals

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Homework Statement


Find the volume of the solid bounded above by the surface z = x^2 + y^2 and below by the
triangular region in the xy-plane enclosed by the lines x = 0 , y = x , and x + y = 8.


Homework Equations


V = ∫∫ Height
Base

The Attempt at a Solution


I first found height, because height = z (upper) - z (lower) = (x^2 + y^2) - 0 = x^2 + y^2

Afterwards, I began solving for the base. I know the base is enclosed by the curves y = x,
y = x - 8, and x = 0. I proceed assuming the region is y-simple, giving me the boundaries:
Base { x ≤ y ≤ 8-x
and 0 ≤ x ≤ 4 }
This sets up the double integral for me, which turns out to be:
4 8-x
∫ { ∫ (x^2 + y^2) dy } dx
0 x

Solving this integral gives me that volume is 1024/3.
Could someone look through my work and see if I made any errors? I'm a beginner at setting up these equations, so I suspect my integral to have a mistake. Thank you in advance.
 

Answers and Replies

  • #2
STEMucator
Homework Helper
2,075
140

Homework Statement


Find the volume of the solid bounded above by the surface z = x^2 + y^2 and below by the
triangular region in the xy-plane enclosed by the lines x = 0 , y = x , and x + y = 8.


Homework Equations


V = ∫∫ Height
Base

The Attempt at a Solution


I first found height, because height = z (upper) - z (lower) = (x^2 + y^2) - 0 = x^2 + y^2

Afterwards, I began solving for the base. I know the base is enclosed by the curves y = x,
y = x - 8, and x = 0. I proceed assuming the region is y-simple, giving me the boundaries:
Base { x ≤ y ≤ 8-x
and 0 ≤ x ≤ 4 }
This sets up the double integral for me, which turns out to be:
4 8-x
∫ { ∫ (x^2 + y^2) dy } dx
0 x

Solving this integral gives me that volume is 1024/3.
Could someone look through my work and see if I made any errors? I'm a beginner at setting up these equations, so I suspect my integral to have a mistake. Thank you in advance.

Hmm I'm getting ##\frac{2048}{3}##. Double your answer.

The way I managed to get that was by starting with ##x = y## and ##x = 8 - y##.

Then ##x = 0## implies that ##y = 0## and ##y = 8##.
 
  • #3
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,559
770

Homework Statement


Find the volume of the solid bounded above by the surface z = x^2 + y^2 and below by the
triangular region in the xy-plane enclosed by the lines x = 0 , y = x , and x + y = 8.


Homework Equations


V = ∫∫ Height
Base

The Attempt at a Solution


I first found height, because height = z (upper) - z (lower) = (x^2 + y^2) - 0 = x^2 + y^2

Afterwards, I began solving for the base. I know the base is enclosed by the curves y = x,
y = x - 8, and x = 0. I proceed assuming the region is y-simple, giving me the boundaries:
Base { x ≤ y ≤ 8-x
and 0 ≤ x ≤ 4 }
This sets up the double integral for me, which turns out to be:
4 8-x
∫ { ∫ (x^2 + y^2) dy } dx
0 x

Solving this integral gives me that volume is 1024/3.
Could someone look through my work and see if I made any errors? I'm a beginner at setting up these equations, so I suspect my integral to have a mistake. Thank you in advance.

Your setup and answer look correct to me.
 
  • #4
79
0
Yup, thank you, I got the correct answer.
 

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