Stewart Calculus Section 10.4

1. 10.4 Area and Length in
Polar Coordinates
Area of a region bounded by the polar curve
= ( )
and by the rays:
= , =
O

2. O
Formula:
= ( )
or
=

3. Ex: Find the area enclosed by one loop of
the four-leaved rose = .

4. Ex: Find the area enclosed by one loop of
the four-leaved rose = .
:

5. Ex: Find the area enclosed by one loop of
the four-leaved rose = .
:
/
=
/

6. Ex: Find the area enclosed by one loop of
the four-leaved rose = .
:
/
=
/
/
=
/

7. Ex: Find the area enclosed by one loop of
the four-leaved rose = .
:
/
=
/
/
=
/
=

8. Ex: Find the are of the region that lies inside
the circle = and outside the
cardioid = + .

9. Ex: Find the are of the region that lies inside
the circle = and outside the
cardioid = + .
:

10. Ex: Find the are of the region that lies inside
the circle = and outside the
cardioid = + .
:
/
= ( )
/
/
( + )
/

11. Ex: Find the are of the region that lies inside
the circle = and outside the
cardioid = + .
:
/
= ( )
/
/
( + )
/
=

12. Arc length of a polar curve = ( )
between = , =
is given by
= +
O

13. Ex: Find the length of the cardioid = +

14. Ex: Find the length of the cardioid = +
= +

15. Ex: Find the length of the cardioid = +
= +
= ( + ) +

16. Ex: Find the length of the cardioid = +
= +
= ( + ) +
=