Section 4-2-B More approximations of area using

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Section 4-2-B More approximations of area using Midpoint formula and Trapezoidal Rule

using the midpoints of rectangles Approximating Area using the midpoints of rectangles Over Estimate: when concave down Under Estimate: when concave up

Midpoint Formula Let n be the number of rectangles used on the interval [a,b]. Then the area approximated using the midpoint is given by: Value of function between f(x)0 and f(x)1. The leftmost endpoint and the second x-value. Value of function between f(x)n-1 and f(x)n. The rightmost endpoint and the second to last x-value Width of each rectangle along the x-axis

Steps for using The Midpoint Formula Graph the function on the interval Determine width of each rectangle and mark on the graph Find the midpoint between each mark and use it to find the function value. (height) Fill in the Midpoint Formula

10) Approximate the area under the curve from x = 0 to x = 6 with 6 rectangles using the midpoints.

11) Approximate the area under the curve from x = 1 to x = 5 with 4 rectangles using the midpoints.

Trapezoidal Approximation for Area Intermediate sides are shared by two trapezoids

Trapezoidal Rule: Let n be the number of trapezoids used on the interval [a,b]. Then the area approximated is given by: Every intermediate value is used twice so multiply by 2 Endpoints only used once Width along x-axis

Trapezoidal Approximations Under Estimate: when concave down Over Estimate: when concave up

12) Approximate the area under the curve from x = 0 to x = 4 with 4 trapezoids.

13) If g(x) is a continuous function, find the area from x = 1 to x = 8 with four trapezoids given the information below. x 1 2 3 6 8 g(x) 4 12 10 When given the information in tabular form, verify the trapezoids have same width before using the Trapezoidal Rule Formula.

16) A metal wire of length 8 centimeters is heated at one end 16) A metal wire of length 8 centimeters is heated at one end. The table gives selected values of the temperature T(x) in degrees Celsius of the wire x cm from the heated end. The function T(x) is decreasing and twice differentiable. Estimate the average temperature of the wire using a trapezoidal sum with four subintervals indicated by the data in the table.

Homework Worksheet: Area Approximations wks 4-2