Mathematics 53
I. Evaluate the following integrals. Z πr 1 + cos 2θ 1. dθ 2 0 Z q √ 2. sin x 2 + cos xdx Z 3.
π 2
Exercise Set 4
υ √ dυ 1+υ υ
Z
y dy (2y + 1)3
5. Z
cos y cos (sin y) dy
−π
√
Z 4.
1
(|2w − 1| + w) dw
6. −1
II. Do as indicated. 1. Consider the shaded region bounded by y = x3 − 2x and y = x2 .
Set-up the definite integrals for: (a) the total area of the region (b) the perimeter of the region
2. Set-up the definite integral for the volume of the solid generated by revolving the region bounded 3 by x2 + y 2 = 1 and y 2 = x about 2 (a) the x-axis using Disk Method (b) the line x = −1 using Cylindrical Shell Method
III. Miscellaneous. 1. At every point of a certain curve, y 00 = x2 −1. Find the equation of the curve if it passes through the point (1, 1) and is there tangent to the line x + 12y = 13. 2. A ball is rolled over a level lawn with initial velocity 25 ft/sec. Due to friction, the velocity decreases at the rate of 6 ft/sec2 . How far will the ball roll? 3. A stone is dropped from the top of a building 484 ft high. The only force acting on it is the force of gravity which is -32 m/s2 . (a) How long will it take the stone to hit the ground? (b) With what speed will it strike the ground? Z 4 4. Express the definite integral x2 dx as a Riemann sum. 2
Z 5. Find the derivative wrt x of F (x) =
0
√
p 1 + t4 dt
x
π π 2π (n − 1) π 6. Compute lim sin + sin + . . . + sin . n→+∞ n n n n 7. Given y = x + 3 in the interval [-3, 0], find the suitable c satisfying the Mean Value Theorem for Integrals.