Indian Institute of Technology Jodhpur
Mathematics 1 (MAL1010)
.
Tutorial 5
2019-20, Semester I
.
Instructor: Dr.Gaurav Bhatnagar
(1) Find the volume of the solid generated by revolving the region between y =
and x-axis, when revolved around the x-axis.
√
x, 0 ≤ x ≤ 4
(2) Find the volume of the solid generated by revolving the region between x = y2 + 1, and the
line x = 3, when revolved around the line x = 3.
(3) Find the volume of the solid generated by revolving the region between y = x2 + 1, and the
line y = −x + 3, when revolved around the x-axis.
(4) The region enclosed by x − axis and the parabola y = 3x − x2 is revolved about the line x = −1.
Find the volume of the solid generated.
(5) Use shell method to find the volume of the solid generated when the region bounded by the
√
curves, y = 4x − x2 , x-axis and the line x = 2 is rotated about the x-axis.
(6) Let R denote the region in the first quadrant bounded by the parabolas y = x2 and y = 2 − x2 .
Determine the volume of the solid generated when the region is rotated about the y-axis.
(7) Find the volume of the solids enclosed by the cylinders x2 + y2 = a2 and x2 + z2 = a2 .
(8) Find the length of the following curves:
(a) y2 + 2y = 2x + 1 from (−1, −1) to (7, 3).
y3
1
(b) x =
+
from x = 0 to x = 4.
3
4y
(c) r = θ, 0 ≤ θ ≤ 1.
(d) x = t2 and y = t3 , (1, 1) to (4, 8).
(9) Find a curve through the point (0,5) whose length integral is L =
∫4 √
1 + 4x2 dx.
1
(10) Derive the formula of length of a curve y = f (x). Extend it when function is represented in
parametric form. (Not to be discussed in tutorial class)
1