1. Area Approximation
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4-C

2. Exact Area Use geometric shapes such as rectangles,
circles, trapezoids, triangles etc…
rectangle
parallelogram
triangle

3. Approximate Area
Midpoint
Trapezoidal Rule

4. Approximate Area
Riemann sums
Left endpoint
Right endpoint

5. Inscribed Rectangles: rectangles remain under the curve
Inscribed Rectangles: rectangles remain under the curve. Slightly underestimates the area.
Circumscribed Rectangles: rectangles are slightly above the curve. Slightly overestimates the area
Left Endpoints

6. Left endpoints:
Increasing: inscribed
Decreasing: circumscribed
Right Endpoints:
increasing: circumscribed, decreasing: inscribed

7. The area under a curve bounded by f(x) and the x-axis and the lines
x = a and x = b is given by
Where
and n is the
number of sub-intervals

8. Therefore:
Inscribed rectangles
Circumscribed rectangles
The sum of the area of the inscribed rectangles is called a lower sum, and the sum of the area of the circumscribed rectangles is called an upper sum

9. Fundamental Theorem of Calculus:
If f(x) is continuous at every point [a, b] and F(x) is an antiderivative of f(x) on [a, b] then the area under the curve can be approximated to be

10. + -

11. Simpson’s Rule:

12. 1) Find the area under the curve
from

13. 2) Approximate the area under
from
With 4 subintervals using inscribed rectangles

14. 3) Approximate the area under
from
Using the midpoint formula and n = 4

15. 4) Approximate the area under the curve
between x = 0 and x = 2
Using the Trapezoidal Rule with 6 subintervals

16. 5) Use Simpson’s Rule to approximate the area under the curve
on the interval using 8 subintervals

17. 6) The rectangles used to estimate the area under the curve on the interval using 5 subintervals with right endpoints will be
Inscribed
Circumscribed
Neither
both

18. 7) Find the area under the curve
on the interval using 4 inscribed rectangles

19. Home Work Worksheet on Area
Use a section header for each of the topics, so there is a clear transition to the audience.
Worksheet on Area