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