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Leibniz’s Rule

If f is continuous function on [a, b] and u(x) and v(x) are differentiable functions of x whose values lie in [a, b] then
 
98344.png
= f(v(x))98338.pngf(u(x))98332.png
 
Inequalities
  1. If at every point x of an interval [a, b] the inequalities g(x) ≤ f (x) ≤ h (x) are fulfilled then98326.png dx98320.pngdx98314.pngdx, a < b
  2. If m is the least value (global minimum) and M is the greatest value (global maximum) of the function f(x) on the interval [a, b] (estimation of an integral). Then m(ba) ≤ 98308.png dxM (ba)
  3. 98302.png98295.png

Different cases of bounded area

  1. The area bounded by the continuous curvey = f(x), the axis of x and the ordinates x= a and x = b (where b > a) is given by
     
    A = 98275.png
98269.png
  1. The area bounded by the straight lines x =a, x = b (a < b) and the curves y = f(x) and y =g(x), provided f(x) ≤ g(x) (axb), is given by
     
    A = 98263.png
98257.png
  1. When two curves y = f(x) and y = g(x) intersect, the bounded area is
     
    A = 98251.png
     
    where a and b are roots of the equation f(x) = g(x).
98245.png
  1. If the curve crosses the x-axis at c, then the area bounded by the curve y = f(x) and the ordinates x = a and x = b (b > a) is given by
     
    98239.png
     
    98232.png
98226.png
  1. The are bounded by y = f(x) and y =g(x), axb, when they intersect at x =c (a, b) is given by
     
    98220.png
     
    or
     
    98213.png
98206.png

Curve tracing

To find the approximate shape of a curve, the following procedure is adopted in order:
  1. Symmetry:
    1. Symmetry about x-axis: If all the powers of “y” in the equation are even then the curve is symmetrical about the x-axis, e.g., y2 = 4ax.
    2. Symmetry about y-axis: If all the powers of “x” in the equation are even then the curve is symmetrical about the y-axis, x2 = 4ay.
    3. Symmetry about both axis: If all the powers of “x” and “y” in the equation are even, the curve is symmetrical about the axis of “x” as well as “y”, e.g., x2 + y2 = a2.
    4. Symmetry about the line y = x: If the equation of the curve remains unchanged on interchanging “x” and “y”, then the curve is symmetrical about the line y = x, e.g., x3 + y3 = 3xy.
  2. Find the points where the curve crosses the x-axis and the y-axis.
  3. Find dy/dx and examine if possible the intervals when f(x) is increasing or decreasing and also stationary points.
  4. Examine what happens to “y” when x ∞ or x –∞.

Some standard area

  1. Area bounded by y = sin x, 0 ≤ xπ and x-axis is 2 sq. units. In fact area of one loop of y = sin x and y = cos x is 2 sq. units
  2. Area bounded by y = logex, y = 0 and x = 0 is 1 sq. unit.
  3. Area of ellipse 98187.png is π ab sq. units.
  4. Area bounded by y2 = 4ax and x2 = 4by, a > 0; b > 0 is = 98175.png sq. units.




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