Stewart Calculus Section 7.3

1. 7.3 Trigonometric
Substitution

2. 7.3 Trigonometric
Substitution
Two Kinds of substitutions
( ( )) ( ) = ( )

3. 7.3 Trigonometric
Substitution
Two Kinds of substitutions
( ( )) ( ) = ( )
Ex1: =

4. 7.3 Trigonometric
Substitution
Two Kinds of substitutions
( ( )) ( ) = ( )
Ex1: = =

5. 7.3 Trigonometric
Substitution
Two Kinds of substitutions
( ( )) ( ) = ( )
Ex1: = =
Ex2: = ·

6. 7.3 Trigonometric
Substitution
Two Kinds of substitutions
( ( )) ( ) = ( )
Ex1: = =
Ex2: = ·
=
− ≤
≤

7. 7.3 Trigonometric
Substitution
Two Kinds of substitutions
( ( )) ( ) = ( )
Ex1: = =
Ex2: = ·
=
= − ≤
≤

8. Expression Substitution Identity
√ =
− − =
− ≤ ≤
√ =
+ + =
− < <
√ =
− − =
≤ <

9. (0,b)
Ex: Find the area.
O (a,0)

10. (0,b)
Ex: Find the area.
O (a,0)
√
= −

11. (0,b)
Ex: Find the area.
O (a,0)
√
= −
=
= ≤ ≤

12. (0,b)
Ex: Find the area.
O (a,0)
√
= −
=
= ≤ ≤
= ( + )

13. (0,b)
Ex: Find the area.
O (a,0)
√
= −
=
= ≤ ≤
= ( + )
=[ + ] =

14. Ex: Find

15. Ex: Find
=
= − ≤
≤

16. Ex: Find
=
= − ≤
≤
= ( )

17. Ex: Find
=
= − ≤
≤
= ( )
= +

18. Ex: Find
=
= − ≤
≤
= ( )
= +
= +

19. Ex: Find

20. Ex: Find
√
=
= < <

21. Ex: Find
√
=
= < <
= | + |+

22. Ex: Find
√
=
= < <
= | + |+
= + +

23. Ex: Find
√
=
= < <
= | + |+
= + +
= + +

24. Ex: Find
+

25. Ex: Find
+ =
− <
= <
+ ·

26. Ex: Find
+ =
− <
= <
+ ·
=

27. Ex: Find
+ =
− <
= <
+ ·
=
=
=

28. Ex: Find
+ =
− <
= <
+ ·
=
=
=
= +

29. Ex: Find
+ =
− <
= <
+ ·
=
=
=
= +
+
= +

30. Ex: Find

31. Ex: Find
= = +

32. Ex: Find
= = +
= ( )( )
=
− <
<

33. Ex: Find
= = +
= ( )( )
=
− <
<
= +

34. Ex: Find
= = +
= ( )( )
=
− <
<
= +
= +

35. Ex: Find
= = +
= ( )( )
=
− <
<
= +
= +
+
= +