integration

Integral calculus
Assignment 10
This assignment covers the material contained
in Chapters 37 to 39.
The marks for each question are shown in
brackets at the end of each question.
Fig. A10.1. The section of the groove is a semicircle
of diameter 50 mm. Given that the centroid
of a semicircle from its base is 4r , use the
3π
theorem of Pappus to determine the volume of
material removed, in cm 3 , correct to 3 significant
figures. (8)
∫
1. Determine (a)
∫
(c)
√
3 t 5 dt
∫
(b)
2
3√
x 2 dx
(2 + θ) 2 dθ (9)
2. Evaluate the following integrals, each correct to
4 significant figures:
∫ π ∫
3
2
( 2
(a) 3 sin 2t dt (b)
x 2 + 1 x 4)
+ 3 dx
(c)
0
∫ 1
0
1
3
dt (15)
e2t 3. Calculate the area between the curve
y = x 3 − x 2 − 6x and the x-axis. (10)
4. A voltage v = 25 sin 50πt volts is applied across
an electrical circuit. Determine, using integration,
its mean and r.m.s. values over the range
t = 0tot = 20 ms, each correct to 4 significant
figures. (12)
5. Sketch on the same axes the curves x 2 = 2y and
y 2 = 16x and determine the co-ordinates of the
points of intersection. Determine (a) the area
enclosed by the curves, and (b) the volume of
the solid produced if the area is rotated one
revolution about the x-axis. (13)
6. Calculate the position of the centroid of the
sheet of metal formed by the x-axis and the part
of the curve y = 5x − x 2 which lies above the
x-axis. (9)
7. A cylindrical pillar of diameter 400 mm has a
groove cut around its circumference as shown in
Figure A10.1
8. A circular door is hinged so that it turns about a
tangent. If its diameter is 1.0 m find its second
moment of area and radius of gyration about the
hinge. (5)
9. Determine the following integrals:
∫
∫ 3lnx
(a) 5(6t + 5) 7 dt (b) dx
x
∫
2
(c) √ dθ (9)
(2θ − 1)
10. Evaluate the following definite integrals:
∫ π
2
(
(a) 2 sin 2t + π ) ∫ 1
dt (b) 3x e 4x2−3 dx
0
3
0
(10)