Department of Mathematics, NIT Calicut
First Semester B.Tech.(CSE/EP) MA1002E Mathematics I, Monsoon 2025-26
.
Tutorial Sheet - 4
Week 4 Portions covered: Area between curves, volume of solid using cross-sectional
area, volume of solid of revolution, area of surface of revolution, improper integrals
1. Sketch the region enclosed by the curves and find its area.
→
(i) y = x2 , y = x, x = 14 , x = 1. (ii) y = cos 2x, y = 0,
(iii) y = sec2 x,
y = 2,
x = ↑ ω4 ,
x = ω4 . (iv) x = sin y,
x = ω4 ,
x = 0,
x = ω2 .
y = ω4 ,
y = 3ω
4 .
2. Assume that f and g are distinct continuous functions on [a, b] and that A denotes the area of
the region bounded by the graphs of y = f (x), y = g(x), x = a, and x = b. Show that,
(i) If f and g di!er by a positive constant c, then A = c(b ↑ a).
(ii) If
! b
a
"
#
f (x) ↑ g(x) dx = 0,
then the graphs of y = f (x) and y = g(x) cross at least once on [a, b].
3. Find the volume of the solid whose base is the region bounded between the curve y = x2 and
the x-axis from x = 0 to x = 2, and whose cross sections taken perpendicular to the x-axis are
squares.
4. Find the volume of the solid that results when the region above the x-axis and below the ellipse
x2 y 2
+ 2 = 1, (a > 0, b > 0)
a2
b
is revolved about the axis.
5. Find the volume of the solid that results when the region enclosed by the given curves is
revolved→about the x-axis.
→
(i) y = 25 ↑ x2 , y = 3 (ii) y = 9 ↑ x2 , y = 0 (iii) x = y, x = y
6. Find the volume of the solid that results when the region enclosed by the given curves is
revolved about the y-axis.
2
2
2
(i) x = csc y, y = ω4 , y = 3ω
4 , x = 0. (ii) y = x , x = y . (iii) x = y , x = y + 2.
7. Find the area of the surface generated by revolving the given curve about the y-axis:
$
(i) x = 9y + 1, 0 ↓ y ↓ 2. (ii) x = y 3 , 0 ↓ y ↓ 1.(iii) x = 9 ↑ y 2 , ↑2 ↓ y ↓ 2.
8. In each part, determine whether the integral is improper, and if so, explain why.
! 5
! 1
! →
! ω/4
dx
↑x
(i)
. (ii)
ln x dx. (iii)
e dx. (iv)
tan x dx.
1 x↑3
0
1
0
! 2
dx
9. Determine values of p for which the integral
is improper.
1 x↑p
! 2
1
10. Check whether the integral,
dx is proper.
x(x
↑ 3)
1
Note: Please inform if there is a typo/logical error.
6