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This tutorial sheet focuses on multiple integrals, covering evaluation techniques, changing the order of integration, and applications in polar, cylindrical, and spherical coordinates. It includes problems on finding areas and volumes using double and triple integrals, along with coordinate transformations. The exercises are designed to enhance understanding of multivariable calculus and its applications in engineering mathematics, providing a comprehensive set of practice problems for students to master these concepts. Useful for university students.
Typology: Exercises
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Tutorial sheet-Multiple Integrals
(a)
0
Z (^) x 2
0
e
y x (^) dydx
(b)
Z (^) a
0
a^2 −y 2
0
dxdy
(c)
Z (^) a
0
Z √ay
0
xydxdy
(d)
Z (^) π
0
Z (^) π
x
sin y
y
dydx
(e)
0
y
x 2 e xy dxdy
0
Z √ 2 −x 2
0
x^2 +y^2
xdzdydx.
Sketch the region of integration and evaluate the integral by expressing the
order of integration dxdydz.
(a) I =
0
Z (^2) −x
x^2
xydxdy
(b.) I =
Z (^2) a
0
Z √ 2 ax
√ 2 ax−x^2
vdxdy.
(a)
0
Z √ 2 x−x 2
0
xdydx p x^2 + y^2
dxdy
(b)
(x − y)
2
x^2 − y^2
dxdy
over the circle x^2 + y^2 ≤ 1.
(c)
Z (^2) a
0
Z (^2) ax−x 2
0
dxdy.
Z Z
D
f (x, y) dxdy,
where f (x, y) = ex
2 and D is the region bounded by the lines y = 0, x = 1
and y = 2x
(a)
− 1
− 2
− 3
dxdydz
(b)
1
0
− 1
x^2 + y^2 + z^2
dxdydz
(c)
R
(x − y − z) dxdydz, where R : 1 ≤ x ≤ 2; 2 ≤ y ≤ 3; 1 ≤ z ≤ 3.
(d)
0
Z (^1) −x
0
Z (^1) −x (^2) −y 2
0
dzdydx
(e)
Z (^) e
0
Z (^) logy
1
Z (^) ex
1
logzdzdxdy
the plane x = 5
(a)
2 , π 4
(b) (1, π, e) (c)
1 , 3 π 2
(a) (1, − 1 , 4) (b)
(c)
Z (^3)
− 3
Z √ 9 −x 2
0
Z (^9) −x (^2) −y 2
0
p x^2 + y^2 dzdydx
by changing to cylindrical coordinates.
(a) (1, 0 , 0) (b)
π 3 ,^
π 4
(c)
5 , π, π 2
(a)
(b) (0, − 1 , −1) (c)
(a) z 2 = x 2
0
Z √ 1 −x 2
0
2 −x^2 −y^2
√ x^2 +y^2
xydzdydx.
