Consider the solid region S that lies under the surface z = x 2 y and above the rectangle R = [ 0 , 2 ] × [ 1 , 4 ] (a) Find a formula for the area of a cross-section of S in the plane perpendicular to the x -axis at x for 0 ⩽ x ⩽ 2 . Then use the formula to compute the areas of the cross-sections illustrated. (b) Find a formula for the area of a cross-section of S in the plane perpendicular to the y -axis at y for 1 ⩽ y ⩽ 4 . Then use the formula to compute the areas of the cross-sections illustrated. (c) Find the value of S.
Consider the solid region S that lies under the surface z = x 2 y and above the rectangle R = [ 0 , 2 ] × [ 1 , 4 ] (a) Find a formula for the area of a cross-section of S in the plane perpendicular to the x -axis at x for 0 ⩽ x ⩽ 2 . Then use the formula to compute the areas of the cross-sections illustrated. (b) Find a formula for the area of a cross-section of S in the plane perpendicular to the y -axis at y for 1 ⩽ y ⩽ 4 . Then use the formula to compute the areas of the cross-sections illustrated. (c) Find the value of S.
Solution Summary: The author calculates the area of a cross-section of S in the plane perpendicular to the x- axis.
Consider the solid region
S
that lies under the surface
z
=
x
2
y
and above the rectangle
R
=
[
0
,
2
]
×
[
1
,
4
]
(a) Find a formula for the area of a cross-section of
S
in the plane perpendicular to the
x
-axis at
x
for
0
⩽
x
⩽
2
.
Then use the formula to compute the areas of the cross-sections illustrated.
(b) Find a formula for the area of a cross-section of
S
in the plane perpendicular to the
y
-axis at
y
for
1
⩽
y
⩽
4
. Then use the formula to compute the areas of the cross-sections illustrated.
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