Real Analysis: Criteria for Integrability

Suppose that the function f:[a,b]->R is integrable and there is a postive number m such that f(x) >= m for all x in [a,b]. Show that the reciprocal function 1/f:[a,b]->R is integrable by proving that for each partition P of the interval [a,b], U(1/f,P) - L(1/f,P) <= 1/m^2[U(f,P) - L(f,P)]

Real Analysis: Criteria for Integrability

Suppose the continuous function f:[a,b]->R has the property that: The integral from c to d f<=0 whenever a<=c

Integration: Cauchy-Schwarz Inequality

Suppose that the functions g:[a,b]-> R are continuous. Prove that: The integral from a to b of gf <= (the square root of the integral from a to b of g^2) multiplied by (the square root from a to b of f^2)

Integration

Note: pi = 3.14...... Prove that (2/pi)x <= sinx <= x if 0 <= x <= pi/2, and use this to prove that: 1 <= the integral from 0 to pi/2 of sinx/x dx <= pi/2

Real Analysis: Geometric Interpretation in Terms of Areas

Note: * = infinite Suppose that the function f:[0,*) -> R is continuous and strictly increasing, and that f:(0,*) -> R is differentiable. Moreover, assume f(0) = 0. Consider the formula: the integral from 0 to x of f + the integral from 0 to f(x) of f^-1 =xf(x) for all x>= 0. How can I provide a geometric interpretation of this formula in terms of areas and then prove this formula. Do I use the Identity Criterion?