Assign-10-H8152.tex 23/6/2006 15: 11 Page 396Integral 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.- Determine (a)
∫
3√
t^5 dt (b)∫
2
√ 3
x^2dx(c)∫
(2+θ)^2 dθ (9)- Evaluate the following integrals, each correct to
4 significant figures:
(a)∫ π
3
03 sin 2tdt (b)∫ 21(
2
x^2+1
x+3
4)
dx(c)∫ 103
e^2 tdt (15)- Calculate the area between the curve
y=x^3 −x^2 − 6 xand thex-axis. (10) - A voltagev=25 sin 50πtvolts is applied across
an electrical circuit. Determine, using integra-
tion, its mean and r.m.s. values over the range
t=0tot=20 ms, each correct to 4 significant
figures. (12) - Sketch on the same axes the curvesx^2 = 2 yand
y^2 = 16 xand 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 thex-axis. (13) - Calculate the position of the centroid of the
sheet of metal formed by thex-axis and the part
of the curvey= 5 x−x^2 which lies above the
x-axis. (9) - A cylindrical pillar of diameter 400 mm has a
groove cut around its circumference as shown in
Fig. A10.1. The section of the groove is a semi-
circle of diameter 50 mm. Given that the centroid
of a semicircle from its base is4 r
3 π, use the
theorem of Pappus to determine the volume of
material removed, in cm^3 , correct to 3 significant
figures. (8)Figure A10.1- 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) - Determine the following integrals:
(a)∫
5(6t+5)^7 dt (b)∫
3lnx
xdx(c)∫
2
√
(2θ−1)dθ (9)- Evaluate the following definite integrals:
(a)∫π
202 sin(
2 t+π
3)
dt (b)∫ 103 xe^4 x(^2) − 3
dx
(10)