126 Ordinary Differential Equati ons
- y = l.5x l.S , y = X 2 3 · , X = 0, X = 1
16. x = siny^2 , x = y/ (y^3 +1), y = 0 , y = 1.2
In exercises 17-22 numerically compute the arc length of the given
curve y = f(x) over the given interval [a, b].
17. f(x ) = x^3 on [O, 1] 18. f(x ) = 1 /x on [.5, 1.5]
19.1
f(x ) = 3x^2 + -
6x^2on [1, 2] 20. f(x ) = ln (1 +ex) on [O, 1.5]
- f(x ) = sinx
x
on [.5, 2] 22. f(x ) = xt an x on [0, 7r/4]
In exercises 23-26 calculate numerically the area of the surface
generated by revolving the given curve y = f(x) over the given
interval [a, b] about (a) the x-axis and (b) the y-axis.
23. f(x ) = 2x^2 on [O, 1] 24. f(x ) = Vx on [1, 2]
- f(x)=sinx on[0,7r]
sin x
2 6. f(x ) = - on [1, 2]
x- Find the surface area and volume of the ellipsoid obtained by revolving
the ellipse x^2 / 16 + y^2 / 25 = 1 (a ) about the x-axis and (b) about the
y-axis. - Find the surface area and volume of the hyperboloid obtained by revolv-
ing the portion of the hyperbola y^2 / 25 - x^2 / 16 = 1 between x = -4
and x = 4 (a ) about the x-axis and (b) about the y-axis.
- Find the surface area and volume of the torus obtained by revolving
the circle (x - 1)^2 + (y - 2)^2 = 1 (a ) about the x-axis and (b) about
the y-axis.
In exercises 30-34 numerically calculate the volume of the solid
obtained by revolving the region bounded by the given set of curves
(a) about the x-axis and (b) about the y-axis.
30. y = sinx^2 , y = 0, x = 0, x = 7r/4
31. y = cosx, y = tan x, x=O
- y = ln x, y = 0, x = 1, x=2
- Y = 3x , y = 3x^3 , x=O