Chapter 27

Find the area between the parabolas \(y^{2}=4 x\) and \(x^{2}=4 y\)

Cylinder: Derive the formula for the volume of a cylinder by rotating the area bounded by the \(x\) axis, the line \(x=h\), and the line \(y=r\) about the \(x\) axis.

Find the area between the parabolas \(y^{2}=2 x\) and \(x^{2}=2 y\)

Cone: Derive the formula for the volume of a right circular cone. Rotate the area bounded by the \(x\) axis, the line \(x=h,\) and the line \(y=r x / h\) about the \(x\) axis.

Sphere: Derive the formula for the volume of a hemisphere by rotating the area bounded by the \(x\) and \(y\) axes and the curve \(x^{2}+y^{2}=r^{2}\) about the \(x\) axis. Double it to get the volume of a sphere.

Approximate methods for evaluating a definite integral include the average ordinate method, the trapezoid rule, and Simpson's rule. Research one or more of these methods and write a short paper on your findings.