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Summary of area formula

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

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Summary of area formula

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  1. Summary of area formula • The area of the region bounded by and is • The area of the region bounded by and is

  2. 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

  3. 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

  4. 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.

  5. 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

  6. 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

  7. 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

  8. 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

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

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