Visvesvaraya National Institute of Technology, Nagpur
Department of Mathematics
Assignment-Integral Calculus: MAL101
R4
1. If f (1) = 2, f 0 is continuous and 1 f 0 (x)dx = 17. What is the value of f (4)?.
1. Length of the curves
(a) Determine the perimeter of one loop of the curve 6ay 2 = x(x − 2a)2
[Ans.
8a
√
]
3
(b) Calculate the distance traveled by the particle P (x, y) after 4 minutes, if the position at any
2
time is given by x = t2 , y = 13 (2t + 1)3/2 .
[Ans. 12]
√
(c) Find the perimeter of the curve r = a(cos θ + sin θ)
[Ans. 2πa]
(d) Determine the perimeter of the curve r = a sin3 ( 3θ )
(e) Find the length of the arc of the parabola r =
√
√
[Ans. ( 2 + ln(1 + 2))2a]
2a
(1+cos θ)
[Ans.
3πa
2 ]
cut-off by its latus rectum.
(f) Find the length of the astroid x = a cos3 t, y = a sin3 t.
[Ans. 6a]
2. Area the curves
(a) Determine the area between the cubic y = x3 and the parabola y = 4x2 .
64
3 ]
3πa2 ]
[Ans.
(b) Calculate the area between the curve y 2 (a + x) = (a − x)3 and its asymptotes.
[Ans.
(c) Find the whole area bounded by the four infinite branches of the tractrix:
x = a cos t + 21 a ln tan2 2t , y = a sin t.
[Ans. πa2 ]
(d) Find the whole area of the curves (i) r = a cos nθ (ii) r = a sin nθ (iii) r = a cos 3θ + b sin 3θ.
2
π(a2 +b2 )
πa2
[Ans. (i) πa
]
4n (ii) 4n , (iii)
4
(e) Let P Q be the common tangent to the two loops of the lemniscate r2 = a2 cos 2θ with pole O.
Find the area bounded by the line P Q and the arcs OP and OQ of the curve.
[Ans.
√
a2
8 (3
3 − 4)]
(f) Compute the area bounded by the x-axis and an arc of the cycloid x = a(t−sin t),y = a(1−cos t).
[Ans. 3πa2 ]
3. Volume and Surfaces of Solid of Revolution
(a) Find the volume of the solid generated by the revolution of an arc of the catenary y = c cosh(x/c)
2
about the x-axis.
[Ans. πc2 (x + 2c sinh(2x/c))]
(b) Determine the volume of solid generated by revolving the plane area bounded by y 2 = 4x and
x = 4 about the line x = 4.
[Ans. 1024
15 π]
(c) Find the volume of the solid generated by revolving the smaller area bounded by the circle
x2 + y 2 = 2 and semicubical parabola y 3 = x2 about the x-axis
[Ans. 52
21 π]
(d) Determine the volume of solid of revolution generated by revolving the curve whose parametric
equation are x = 2t + 3, y = 4t2 − 9 about the x-axis for t1 = − 32 , t2 = 32 .
[Ans. 1296π]
(e) The arc of the cardioid r = a(1 + cos θ) included between θ = −π/2 and θ = π/2 is rotated
3
about the line θ = π/2. Find the volume of the solid of revolution.
[Ans. πa4 (16 + 5π)]
x
−x
(f) Find the volume of the solid generated by the revolution of the catenary y = a2 (e a + e a ) about
3
2
the x-axis between x = 0 and x = π2 .
[Ans. πa8 (e2b/a − e−2b/a ) + πa2 b ]
(g) Find the volume of the solid obtained by rotating about the y-axis, the region bounded by
y = 2x2 − x3 and y = 0.
[Ans. 16π
5 ]
(h) Find the volume of the solid obtained by rotating about the x-axis, the region bounded by
x = 1 + y 2 , x = 0,y = 1, y = 2.
[Ans. 21π
2 ]
(i) Determine the surface of a paraboloid generated by revolution about the x-axis of an arc of the
parabola
y 2 = 2px, which corresponds to the variation of x from x = 0 to x = a.
[Ans.
√
2π p
3/2
3/2
− p ]]
3 [(2a + p)
(j) The astroid x = a sin3 t,y = a cos3 t is revolved about the x-axis.Find the surface of the solid of
2
revolution.
[Ans. 12πa
5 ]
(k) The arc of the parabola y = x2 from (1, 1) to (2, 4) is rotated about the y-axis.Find√the area
√ of
the resulting surface.
[Ans. π6 [17 17 − 5 5]]
(l) Find the volume of the solid obtained by revolving the cissoid y 2 (2a − x) = x3 about its
asymptote.
[Ans. 2π 2 a3 ]
(m) Find the volume of the solid obtained by revolving the area bounded by the curve y 2 = x and
the line y = 4 about the line x = 2.
[Ans. 128π
3 ]
(n) A circular arc revolves about its chord. Find the area of the surface generated, when 2α is the
angle subtended by the arc at the center.
[Ans. 4πa2 (sin α − α cos α)]
4. Beta and Gamma functions.
Z 2
−1
(a) Evaluate
(8 − x3 ) 3 dx.
0
1
Z
(b) Evaluate
0
x(m−1) (1−x)(n−1)
dx.
(a+bx)(m+n)
Z
1
√
(c) Show that
0
Z
(d) Show that
Z
(e) Evaluate
(f) Show that
π
2
0
∞
1
dx
(1−xn )
=
p
( sin θ)dθ
Z
1
Γ( 12 )Γ( n
)
.
1
nΓ( n
+ 12 )
π
2
√
1
dθ
( sin θ)
=
β(p,q)
p+q ,
0
= π.
x8 (1−x6 )
dx.
(1+x)24
0
β(p,q+1)
q
=
β(p+1,q)
p
where p and q are positive.
5. Differentiation under integral sign.
Z π
ln(1+sin α cos x)
dx.
(a) Evaluate
cos x
Z0 ∞
2 a2
(b) Evaluate
e−(x + x2 ) dx.
Z0 ∞
arctan(ax)
(c) Evaluate
dx.
x(1+x2 )
0
Z
π
2
2
sin x
( 1+y
)dx.
sin2 x
0
Z ∞
Z ∞
√
2)
2
π
(−x
(e) Using the result
e
dx = 2 , evaluate
e(−x ) cos(αx)dx.
0Z
0
∞
√
(−α2 )
2)
π
(−x
Hence deduce that
xe
sin(αx)dx = 4 αe 4
(d) Evaluate
0