In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)
∫ from 0 to 2 of sin(x + 1) dx

88. The region in Exercise 87 is revolved about the x-axis to generate a solid.

b. Show that the inner and outer surfaces of the solid have infinite area.

Consider the region bounded by the graphs of

y = arctan(x), y = 0, and x = 1.

b. Find the volume of the solid formed by revolving this region about the y-axis.

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from 1 to 2 of x dx

89. Consider the infinite region in the first quadrant bounded by the graphs of

y = 1 / x², y = 0, and x = 1.

b. Find the volume of the solid formed by revolving the region (i) about the x-axis.

Consider the region bounded by the graphs of y = sin⁻¹(x), y = 0, and x = 1/2.

b. Find the centroid of the region.

In Exercises 11–22, estimate the minimum number of subintervals needed to approximate the integrals with an error of magnitude less than 10^-4 by (b) Simpson’s Rule. (The integrals in Exercises 11–18 are the integrals from Exercises 1–8.)

∫ from -1 to 1 of (x² + 1) dx