A REA A PPROXIMATION 4-B

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

Approximate Area Midpoint Trapezoidal Rule

Approximate Area Riemann sums Left endpoint Right endpoint

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

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

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

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

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

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Simpson’s Rule:

1) Find the area under the curve from

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

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

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

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

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

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

H OME W ORK Worksheet on Area