84.a. Find the center of mass of a thin plate of constant density covering the region between the curve y=1/√x and the x-axis from x=1 to x=16.

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

3. a. arcsin(-1/2)

Evaluate the integrals in Exercises 67–74 in terms of

a. inverse hyperbolic functions.

67. ∫(from 0 to 2√3)dx/√(4+x²)

Suppose that the function g and its derivative with respect to x have the following values at x=0, 1, 2, 3, and 4.

Assuming the inverse function g^(-1) is differentiable, find the slope of g^(-1)(x) at

a. x=1

Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.

7. a. sec^(-1)(-√2)

31. The incidence of a disease (Continuation of Example 4.) Suppose that in any given year the number of cases can be reduced by 25% instead of 20%.

a. How long will it take to reduce the number of cases to 1000?

143.

a. Show that ∫ ln(x) dx = x ln(x) − x + C.