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Tutorial sheet on triple integral
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Worksheet No. 04. Lecturer: Mr. T. Sinyangwe.
(1) Determine the region of integration first and then evaluate the integral ∫ (^1)
0
dx
∫ (^1) −x
0
dy
∫ (^5) −x 2
4 x^2
(x − y + 1)dz
Considering the figure below;
(2) Evaluate the iterated integral ∫ (^1)
0
∫ (^) z
0
∫ (^) y
0
xyzdxdydz
(3) The figure below shows the region of integration for the integral
∫ (^1)
0
√ 5
∫ (^1) −y
0
f (x, y, z)dzdydx
Rewrite this integral as an equivalent iterated integral in the five other orders.
(4) Evaluate the triple integral
(i)
E
zdV , where E is bounded by the planes x = 0, y = 0, z = 0, y + z = 1 and
x + z = 1, as shown in the figure below;
(7) Evaluate by converting to cylindrical polar coordinates E
2 xdV , where E is the
region under the plane 2x + 3y + z = 6 that lies in the first octant as shown below;
(8) Determine the volume of the region by converting to cylindrical polar form of the region that lies behind the plane x + y + z = 8 and in front of the region in the
xy-plane that is bounded by z =
y & z =
y as shown below;
(9) Evaluate by converting to cylindrical polar coordinates
E
3 x^2 + 3z^2 dV where
E is the solid bounded by y = 2x^2 + 2z^2 and the plane y = 8 as shown below;
(10) Evaluate the integral
2
− 1
1
4 x^2 y − z^3 dzdydx.
(11) Evaluate E
4 xydV where E is the region bounded by z = 2x^2 + 2y^2 − 7 and z = 1
as shown below;
(13) Evaluate E
zdV where E is the region between the two planes x + y + z = 2 and
x = 0 and inside the cylinder y^2 + z^2 = 1 as shown below;
(14) Use triple integral to determine the volume of the region below z = 6 − x, above
z = −
4 x^2 + 4y^2 inside the cylinder x^2 + y^2 = 3 with x ≤ 0 as shown below;
(15) Evaluate the following integral by first converting it to an integral in cylindrical co-
ordinates ∫ √ 5
0
−
√ 5 −x^2
∫ (^9) − 3 x (^2) − 3 y 2
x^2 +y^2 − 11
2 x − 3 ydzdydx.
Triple integrals in Spherical coordinates
(16) Evaluate
E
10 xz + 3dV where E is the region portion of x^2 + y^2 + z^2 = 16 with
z ≥ 0 as shown below;
(17) Evaluate
E
x 2
as shown below;
(19) Evaluate E
x 2 dV where E is inside both x^2 + y^2 + z^2 = 36 and z = −
3 x^2 + 3y^2
as shown below;
(20) Evaluate the following integral by first converting to an integral in spherical coordi-
nates, ∫ (^0)
− 1
∫ √ 1 −x 2
−
√ 1 −x^2
∫ √ 7 −x (^2) −y 2
√ 6 x^2 +6y^2
18 ydzdydx.
♣ End of Worksheet No. 04 ♣
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(^1) ”You never know how strong you are until being strong is the only choice you have”∼(Bob Marley)
