SECTION 4-2-B More area approximations. Approximating Area using the midpoints of rectangles.

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Presentation transcript:

SECTION 4-2-B More area approximations

Approximating Area using the midpoints of rectangles

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 y 0 and y 1. The leftmost endpoint and the second x-value. Value of function between y n-1 and y n. The rightmost endpoint and the second to last x-value Width of each rectangle along the x-axis

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

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 Fill in the Midpoint Formula Steps for using 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 = 4 with 4 rectangles using the midpoints.

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

Trapezoidal Approximations Under Estimate: when concave down Over Estimate: when concave up Intermediate sides used for two trapezoids

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. x12368 g(x) When given the information in tabular form, verify the trapezoids have same width before using the Trapezoidal Rule Formula.

Simpson’s Rule: Let n be the number of subintervals (must be even) used on the interval [a,b]. Then the area approximated is given by: width along x-axis Endpoint only used once Every intermediate value alternates (+4) then (+2)

Simpson’s Rule Approximations Under Estimate: when concave down Over Estimate: when concave up

14) Approximate the area under the curve from x = 0 to x = π with n = 4 using Simpson’s rule

15) Which method will overestimate and which will underestimate the area under the curve on the given interval Increasing and Decreasing Concave up and Concave Down

Homework Worksheet: Area Approximations wks 4-2