A REA A PPROXIMATION 4-E Riemann Sums

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

Approximate Area: Add the area of the Rectangles Midpoint

Approximate Area: Add the area of the Rectangles 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 exact 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

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) 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

6) Find approximate the area under the curve on the interval using right hand Riemann sum with 4 equal subdivisions

7) Approximate by using 5 rectangles of equal width and an Upper Riemann Sum

H OME W ORK Area Approximations worksheet