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Unformatted text preview: M10360 Exam 2: Review Guide Chapter 6 6.4: First Order Linear Differential Equations • Step 1: Write the problem as y + P ( x ) y = Q ( y ). i.e., isolate the y and then factor the y out of any expression it is in. The stuff attached to the y is P ( x ) and the rest is called Q ( x ). Both P ( x ) and Q ( x ) must be in terms of x only. • Step 2: Calculate Z P ( x ) dx with C = 0. Then find the integrating factor u ( x ) = e R P ( x ) dx . • Step 3: The answer is given by y = 1 u ( x ) Z Q ( x ) u ( x ) dx . Remember to include the constant C inside the parentheses. Chapter 7 7: Basic Idea: Working with Slices • Draw a rough sketch of the situation. • Pick a generic looking slice and label all the information about that slice that is relevant for the problem. This usually includes dx (or dy ) and a coordinate ( x, y ). • Use the labeled information above to find out the desired property about the slice, like area, volume, arc length, surface area, work, mass, moment, or fluid force. • Integrate over the correct bounds of the formula found for the small slice. Think of this as “adding” up all of the slices in your bounds. • Note: Sometimes you may have to cut your region into two or more regions so that the slice really encapsulates what is happening over a given region. (i.e finding the work of rolling up half of a chain on a winch requires you to treat the top half of the chain differently from the bottom half). 7.1: Area Between Two Curves • Slices are rectangles. Area of slice is (width of slice) · (height of slice). • The width of the slice is typically dx or dy depending on what your slices look like. Remember if your slice changes a little in the x-direction, use dx and vice-versa for y . • The height of the slice is usually (top curve -bottom curve) or (right curve-left curve), but you must think your way through any given problem....
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