25 Jun 1998

### lane Areas

1. The area bounded by the curve y = f(x), the vertical lines x = a, x = b, and the x-axis is given by
 A = ób õa y dx = ób õa f(x) dx.

1. If the equation of the curve is given in parametric form  x = g(t), y = h(t), then
 A = ób õa y dx = óv õu h(t)g'(t) dt
where u and v are the values of t when x = a and b respectively.

1. The area bounded by the curves y = f(x) and y = g(x) is given by
 A = ób õa [f(x) - g(x)] dx
where a and b are the x-coordinates of the points of intersections.

1. If the curve y = f(x) crosses the x-axis at x = m, (a < m < b), then the area bounded by the curve, x = a, x = b and the x-axis is given by
 A = óm õa f(x) dx + ób õm -f(x) dx.

1. The area bounded by the y-axis, the lines y = c, y = d, and the curve y = f(x) is given by
 A = ód õc x dy.

### olumes of Revolution

1. When the curve y = f(x) is rotated about the x-axis through 4 right angles, the volume of revolution between x = a and x = b is given by
 V = ób õa py2 dx.

1. When the curve x = g(y) is rotated about the y-axis through 4 right angles, the volume of revolution between y = c and y = d is given by
 V = ód õc px2 dx.

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