1000 Solved Problems in Modern Physics

26 1 Mathematical Physics

1.46 Show that:
∫ 4

2

2 x+ 4

x^2 − 4 x+ 8

dx=ln 2+π

1.47 Find the area included between the semi-cubical parabolay^2 =x^3 and the
linex= 4

1.48 Find the area of the surface of revolution generated by revolving the hypocy-
cloidx^2 /^3 +y^2 /^3 =a^2 /^3 about thex-axis.

1.49 Find the value of the definite double integral:
∫a

0

∫√a (^2) −x 2
0
(x+y)dydx
1.50 Calculate the area of the region enclosed between the curvey = 1 /x,the
curvey=− 1 /x, and the linesx=1 andx=2.
1.51 Evaluate the integral:
∫
dx
x^2 − 18 x+ 34
1.52 Use integration by parts to evaluate:
∫ 1
0
x^2 tan−^1 xdx
[University of Wales, Aberystwyth 2006]
1.53 (a) Calculate the area bounded by the curvesy=x^2 +2 andy=x−1 and
the linesx=−1 to the left andx=2 to the right.
(b) Find the volume of the solid of revolution obtained by rotating the area
enclosed by the linesx= 0 ,y= 0 ,x =2 and 2x+y=5 through 2π
radians about they-axis.
[University of Wales, Aberystwyth 2006]
1.54 Consider the curvey=xsinxon the interval 0≤x≤ 2 π.
(a) Find the area enclosed by the curve and thex-axis.
(b) Find the volume generated when the curve rotates completely about the
x-axis.

1.2.8 OrdinaryDifferentialEquations......................

1.55 Solve the differential equation:

dy

dx

=

x^3 +y^3

3 xy^2