Question

Find the area of the triangular region with vertices : (0,1,0), (2, 1, -1), (1, 2, -1)

Answer #1

Let T be the triangular region with vertices (1,0,0)(1,0,0),
(0,1,0)(0,1,0), and (0,0,1)(0,0,1) oriented with upward-pointing
normal vector.
A fluid flows with constant velocity field v=4i+6j m/sv=4i+6j
m/s. Calculate:
(a) The flow rate through T
(b) The flow rate through the projection of T
onto the xyxy-plane [the triangle with vertices (0,0,0)(0,0,0),
(1,0,0)(1,0,0), and (0,1,0)(0,1,0)]
Assume distances are in meters.
(a) ∬Sv⋅dS=∬Sv⋅dS=
(b) ∬Sv⋅dS

Find the mass of the triangular region with vertices (0, 0), (1,
0), and (0, 5), with density function ρ(x,y)=x2+y2

Find the mass of the triangular region with vertices (0, 0), (3,
0), and (0, 5), with density function (x,y)=x^2+y^2.

Find the absolute maximum and minimum values of
f(x,y)=2x^2+y^2-xy^2 on the triangular region shown with vertices
(0,0), (0,4) and (4,4).

Find the absolute maximum value of the function
f(x,y)=x2-4xy+y3+4y on the triangular region
with vertices (-1,-1), (7,-1) and (7,7).

2. Volume
(a) Compute volume of the solid whose base is a triangular
region with vertices (0,0), (1,0), and (0,1), and whose
cross-sections taken perpendicular to the y -axis are equilateral
triangles.
(b) Compute the volume of the solid formed by rotating the
region between the curves x=(y-3)^2 and x = 4 about the line y
=1

Integrate the function f over the given region
f(x,y) =xy over the triangular region with vertices (0,0) (6,0)
and(0,9)

A thin plate covers the triangular region of the xy-plane with
vertices (0,0), (1,1), and (−1,1). (Coordinates measured in
cm.)
(a) Find the mass of the plate if its density at (x,y) is
sin(y^2) kg/cm^2 .
(b) Find the mass of the plate if its density at (x,y) is
sin(x^2) kg/cm^2 .

Find the absolute maximum and minimum values of f(x,y)=4xy+x^2
on the triangular region D in the xy plane with the vertices (4,0)
(0,3) and (2,4)

Find the absolute minimum and absolute maximum of
f(x,y)=10−3x+8y
on the closed triangular region with vertices (0,0),(8,0) and
(8,12).
List the minimum/maximum values as well as the point(s) at which
they occur. If a min or max occurs at multiple points separate the
points with commas.

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