Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.
83. Find the area of the region.

Centroid: Find the centroid of the region bounded by the x-axis, the curve y = csc x, and the lines x = π/6, x = 5π/6.

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ θ cos(πθ) dθ

Volume: Find the volume of the solid generated by revolving the region in Exercise 45 about the x-axis.

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (dx / ((2x + 1)√(4x + 4x²)))

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (2^(√y) dy) / 2√y

Expand the quotients in Exercises 1–8 by partial fractions.

z / (z³ - z² - 6z)