Centroid of a region
Find the centroid of the region in the plane enclosed by the curves y = ±(1 − x²)^(-1/2) and the lines x = 0 and x = 1.

Finding volume

Let R be the "triangular" region in the first quadrant that is bounded above by the line y = 1, below by the curve y = ln x, and on the left by the line x = 1.

Find the volume of the solid generated by revolving R about

a. the x-axis.

Evaluate the limits in Exercise 9 and 10 by identifying them with definite integrals and evaluating the integrals.

lim (n → ∞) Σ (from k=1 to n) ln √(1 + k/n)

Finding volume

The infinite region bounded by the coordinate axes and the curve y = −ln x in the first quadrant is revolved about the x-axis to generate a solid. Find the volume of the solid.

Evaluate the integrals in Exercises 1–6.

∫ dt / (t - √(1 - t²))

Finding volume

The region in the first quadrant enclosed by the coordinate axes, the curve y = e^x, and the line x = 1 is revolved about the y-axis to generate a solid. Find the volume of the solid.

Use the substitution z = tan(θ/2) to evaluate the integrals in Exercises 41 and 42.

∫ csc θ dθ