Summary of area formula The area of the region bounded by and is

Summary of volume formula The volume of the solid obtained by rotating about x-axis the region enclosed by y=f(x), x=a, x=b and x-axis, is The volume of the solid obtained by rotating about y-axis the region enclosed by x=f(y), y=c, y=d and y-axis, is

Volume by cylindrical shells The volume of the solid obtained by rotating about y-axis the region enclosed by y=f(x), x=a, x=b and x-axis, is The volume of the solid obtained by rotating about x-axis the region enclosed by x=f(y), y=c, y=d and y-axis, is

Example Ex. Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line: (1) about (2) about (3) about x-axis Sol.

Physical application: work Problem: Suppose a force f(x) acts on an object so that it moves from a to b along the x-axis. Find the work done by the force f(x). Solution: take any element [x,x+dx], the work done in moving the object from x to x+dx is so the total work done is

Example Ex. A force of 40N is required to hold a spring that has been stretched from its natural length of 10cm to 15cm. How much work is done in stretching the spring 3cm further? Sol. By Hooke’s Law, the spring constant is k=40/(0.15-0.1)=800. Thus to stretch the spring from the natural length 0.1 to 0.1+x, the force will be f(x)=800x. So the work done in stretching it from 0.15 to 0.18 is

Example Ex. A container which has the shape of a half ball with radius R, is full of water. How much work required to empty the container by pumping out all of the water? Sol. We first set up a coordinate system: origin is the center of the ball and vertical downward line is x-axis. For any take an infinitesimal element [x,x+dx]. The water corresponding to this small part has volume To pump out this part of water, the work required is Therefore total work is

Average value of a function The average value of f on the interval [a,b] is defined by The Mean Value Theorem for Integrals If f is continuous on [a,b], then there exists a number c such that

Homework 16 Section 6.2: 14, 18 Section 6.3: 7, 14 Page 470: 3