1. Find the area of the largest trapezium that can be inscribed in a circle of radius 1 and whose base is a diameter of the circle.
2. Use Newton’s method to find the absolute maximum value (up to 6 decimal places) of the function f (x) = x sin x; 0≤x≤π.
3. Express the following as a definite integral ∫(0,1) f(x) dx and find its value.
lim n→∞ Σ(i=1; n) i/i²+n².
4. Evaluate the following integrals
- ∫(0,1) [1/√(x+1)+√x]dx,
- ∫(3,2) [1/x(x^4+1)]dx.
5. Let R be the region bounded by the curve y = sin x² ; x = 0; x = √π and the x-axis. Find the volume when R is rotated 2π radians about the y-axis.