Quote of the Day: "Nothing was ever achieved without enthusiasm." - Emerson Objectives: The student will integrate integrals involving sums and differences of squares. 1. Collect homework. 2. Recall:
3. Given two sides of a right triangle, a and u, what are the three possibilities that exist? Draw the triangles and label the third side in each.
We will use these three triangles any time that we have an integral involving the sum or difference of two squares. We will use the letter a to represent constants and the letter u to represent variables. This requires no memorization, since you just set up the triangle so that the third side corresponds to what you are looking for (a2 - u2, u2 - a2, or a2 + u2). Here is how you use the triangle:
What we are doing is taking an integral involving the sum or the difference of two squares which we can not integrate (because it is in the denominator or because it is under a square root), and making an appropriate trigonometric substitution so that we obtain an integral of just one term. We can then integrate it or take its square root. 4. Examples:
5. Assignment: p. 535 (1, 4, 5)