26
1 Mathematical Physics
1.46 Show that:
4
2
2x + 4
x 2 − 4x + 8
dx = ln 2 + π
1.47 Find the area included between the semi-cubical parabola y
2
= x
3 and the
line x = 4
1.48 Find the area of the surface of revolution generated by revolving the hypocycloid x
2/3
+ y
2/3
= a
2/3 about the x-axis.
1.49 Find the value of the definite double integral:
a
0
√
a 2 −x 2
0
(x + y) dy dx
1.50 Calculate the area of the region enclosed between the curve y = 1/x, the
curve y = −1/x, and the lines x = 1 and x = 2.
1.51 Evaluate the integral:
dx
x 2 − 18x + 34
1.52 Use integration by parts to evaluate:
1
0
x
2 tan
−1 x dx
[University of Wales, Aberystwyth 2006]
1.53 (a) Calculate the area bounded by the curves y = x
2
+ 2 and y = x − 1 and
the lines x = −1 to the left and x = 2 to the right.
(b) Find the volume of the solid of revolution obtained by rotating the area
enclosed by the lines x = 0, y = 0, x = 2 and 2x + y = 5 through 2π
radians about the y-axis.
[University of Wales, Aberystwyth 2006]
1.54 Consider the curve y = x sin x on the interval 0 ≤ x ≤ 2π .
(a) Find the area enclosed by the curve and the x-axis.
(b) Find the volume generated when the curve rotates completely about the
x-axis.
1.2.8 Ordinary Differential Equations
1.55 Solve the differential equation:
dy
dx
=
x
3
+ y
3
3x y 2
1 Mathematical Physics
1.46 Show that:
4
2
2x + 4
x 2 − 4x + 8
dx = ln 2 + π
1.47 Find the area included between the semi-cubical parabola y
2
= x
3 and the
line x = 4
1.48 Find the area of the surface of revolution generated by revolving the hypocycloid x
2/3
+ y
2/3
= a
2/3 about the x-axis.
1.49 Find the value of the definite double integral:
a
0
√
a 2 −x 2
0
(x + y) dy dx
1.50 Calculate the area of the region enclosed between the curve y = 1/x, the
curve y = −1/x, and the lines x = 1 and x = 2.
1.51 Evaluate the integral:
dx
x 2 − 18x + 34
1.52 Use integration by parts to evaluate:
1
0
x
2 tan
−1 x dx
[University of Wales, Aberystwyth 2006]
1.53 (a) Calculate the area bounded by the curves y = x
2
+ 2 and y = x − 1 and
the lines x = −1 to the left and x = 2 to the right.
(b) Find the volume of the solid of revolution obtained by rotating the area
enclosed by the lines x = 0, y = 0, x = 2 and 2x + y = 5 through 2π
radians about the y-axis.
[University of Wales, Aberystwyth 2006]
1.54 Consider the curve y = x sin x on the interval 0 ≤ x ≤ 2π .
(a) Find the area enclosed by the curve and the x-axis.
(b) Find the volume generated when the curve rotates completely about the
x-axis.
1.2.8 Ordinary Differential Equations
1.55 Solve the differential equation:
dy
dx
=
x
3
+ y
3
3x y 2