Plane Areas Volumes of Revolution

lane Areas

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. 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.

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.

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.

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

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.

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.