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Rough meaning of grades: 29-34 is A 23-28 is B 17-22 is C 11-16 is D Regarding the quiz--if you do every homework problem that was assigned, you'll have a severe case of deja vu on the quiz! On the exam, we do not restrict ourselves like this, but you get to have a sheet of paper. |

The first homework problem is to compute

Your first idea might be to do some sort of substitution, e.g., , but is nowhere to be seen and this simply doesn't work. Likewise, integration by parts gets us nowhere. However, a technique called ``inverse trig substitutions'' and a trig identity easily dispenses with the above integral and several similar ones! Here's the crucial table:

Expression | Inverse Substitution | Relevant Trig Identity |

or |

Inverse substitution works as follows. If we write , then

If is a function, then you can even use inverse substitution for a definite integral. The limits of integration are obtained as follows.

To help you understand this, note that as varies from to , the function varies from to , so is being integrated over exactly the same values. Note also that (5.3.2) once again illustrates Leibniz's brilliance in designing the notation for calculus.

Let's give it a shot with (5.3.1). From the table we use the inverse substition

Wow! That was like magic. This is really an amazing technique. Let's use it again to find the area of an ellipse.

Let's try inverse substitution. The table above suggests using , so . We get

(5.6) | ||

(5.7) | ||

(5.8) | ||

(5.9) |

Thus the area is

One other important technique is to use completing the square.

Of course, we

Here we use that . Also, to compute , we draw a right triangle with side lengths and , and hypotenuse .

[[Draw triangle with sides and and hypotenuse . Then

Back to the integral, we have

William Stein 2006-03-15