MH1810: Find the Area of the Largest Trapezium that can be Inscribed in a Circle of Radius 1: Math 1 Assignment, NTU, Singapore

Questions:

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.