How do you use the triple integral to find the volume of the solid bounded by the surface #z=sqrt y# and the planes x+y=1, x=0, z=0?
4/15 cubic units
The surface of this solid comprises three planar sides (horizontal) z =
0 and (vertical) x = 0 and x + y +1 and a part, in the first octant, of the
symmetrical about the xy-plane.
Now, the volume of this solid in the first octant is
from 0 to 1.
Integrating with respect to z first, in order,,
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first and key, because of the shape of
so we are in the first octant for all of this.
the further constraint is the plane x + y = 1
this is the best drawing i can muster
the yellow bit is the area over which we are integrating z(x,y)
but as a triple integral you would write:
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To find the volume of the solid bounded by the surface (z = \sqrt{y}) and the planes (x+y=1), (x=0), and (z=0), we set up the triple integral over the region bounded by these surfaces.
The region of integration in this case is a triangular region in the xy-plane, bounded by the lines (x=0), (x+y=1), and the y-axis.
We can express the limits of integration as follows:
For (z), it ranges from 0 to (\sqrt{y}). For (y), it ranges from 0 to 1. For (x), it ranges from 0 to (1-y).
The integral setup is:
[ \iiint_V dz , dy , dx ]
with limits of integration:
[ 0 \leq z \leq \sqrt{y}, \quad 0 \leq y \leq 1, \quad 0 \leq x \leq 1 - y ]
Finally, we integrate over these limits to find the volume:
[ \int_0^1 \int_0^{1-y} \int_0^{\sqrt{y}} dz , dx , dy ]
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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