Tutorial Sheet Module 4
BMAT101L - Calculus
1. Evaluate the following :
Z 3Z 2
Z 1 Z √x
−2
(i)
(x + y) dxdy (ii)
xy(x + y)dxdy
4
1Z
0 Z πx Z
Z
Z 2Z y
π
a cos θ
sin θ
2
dxdy
(iii)
r sin θdrdθ (iv)
rdθdr (v)
2
2
0
0
0
0
1
0 x +y
Z a Z √a2 −x2
dxdy.
2. Sketch the region of integration and evaluate
√
0
ax−x2
3. Evaluate the following by Change the order of integration √
Z 1 Z 2−y
Z a Z 2a−x
Z a Z a2 −y2
(i)
xydxdy (ii)
ydxdy
xydydx (iii)
x2
0
y
0
a−y
0
a
Z b Z a/b√b2 −y2
Z 1 Z √y
x
(iv)
xydxdy (v)
dxdy
2
x + y2
0
0
0
y
4. Find the area of a circle of radius ’ a ’ by double integration in polar coordinates.
5. Evaluate the following integrals
by changing variable to polar coordinates
Z ∞Z y
Z a Z √a−x2
Z aZ a
x2
(i)
dxdy (ii)
dydx
(iii)
dxdy
√
3/2
0
0
−a − a−x2
0
y (x2 + y 2 )
Z 2a Z √2ax−x2
Z aZ a
x2
p
(iv)
dxdy (v )
x2 + y 2 dydx
x2 + y 2
0
y
0
0
6. Find, by double integration, the area between the parabola y 2 = 4ax and the
line y = x.
7. Find the area common to y 2 = 4ax and x2 = 4ay using double integration.
8. Find the smaller area bounded by y = 2 − x and x2 + y 2 = 4.
9. Find the area of the cardioid r = a(1 + cos θ) by using double integration.
10. Find the area which is inside the circle r = 3a cos θ and outside the cardioid
r = a(1 + cos θ).
BMAT101L - Calculus
Triple Integral Problems:
11. Evaluate the integrals:
Z 1Z 1Z 1
Z √2 Z 3y Z 8−x2 −y2
dzdxdy
(i)
(x2 + y 2 + z 2 )dzdxdy (ii)
x2 +3y 2
0
0
0
0
0
Z ln 2 Z x Z x+y
e(x+y+z) dzdxdy
(iii)
0
0
0
Z 1 Z √1−x2 Z √1−x2 −y2
1
p
(iv)
dzdxdy
2
1 − x − y2 − z2
0
0
0
12. Evaluate
ZtheZintegrals:
Z
x
ln a
x+y
e(x+y+z) dzdxdy
(i)
0
0
0
Z a Z b 1− x Z c 1− x − y
a
a
b
(ii)
Z0 a Z0 b Z c
x2 zdzdydx
0
2
2
2
(x + y + z )dxdydz
(iii)
0
0
Z 1 Z z Z x+z
(iv)
(x + y + z)dxdydz
−1
0
0
x−z
13. Express the region x ≥ 0, y ≥ 0, z ≥ 0, x2 + y 2 + z 2 ≤ 4 by triple integration and
evaluate the same.
14. Find the volume bounded by the cylinder x2 + y 2 = 9 and the planes y + z = 9 and
z = 0.
15. Find the volume region bounded by the surfaces y 2 = x, x2 = y, the planes z = 0
and z = 3.
16. Find the volume common to the cylinders x2 + y 2 = a2 and x2 + z 2 = a2 .
ZZZ
1
p
dxdydz over the region R bounded by the sphere
17. Evaluate
2
1 − x − y2 − z2
R
x2 + y 2 + z 2 = 1.
18. Find the volume cut off from the sphere x2 +y 2 +z 2 = a2 by the cylinder x2 +y 2 = ax.
ZZZ
19. Evaluate the integral
(x2 + y 2 + z 2 )dxdydz over the region R of space defined
R
by x2 + y 2 ≤ 1 and 0 ≤ x ≤ 1 by changing into cylindrical co-ordinates.
Z 2π Z π Z (1−cos ϕ)/2
20. Evaluate the integral in spherical coordinates
ρ2 sin ϕ dρ dϕ dθ.
0
ZZZ
21. Evaluate the integral
0
0
2
2
2
|xyz|dxdydz over the region R ellipsoid xa2 + yb2 + zc2 ≤ 1.
R
( Hint: Let x = au, y = bv, z = cw. Then integrate over an appropriate region in
uvw-space.)
2