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    Integration using left, right, trapezoid, and midpoint rules

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    Definite integrals
    1. Show geometrically why int sqrt (2-x^2) dx = pi/4 + 1/2
    2. Approximate int sqrt (2-x^2) dx for n = 5 using the left, right, trapezoid, and midpoint rules. Compute the error in each case using the answer to question 1 to compare the errors

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    Solution Preview

    Please see the attached file.

    Show geometrically why .

    This is an equation of a circle with radius sqrt 2 as shown above.

    The given integral is the area of the circle from x = 0 to x = 1 in the first quadrant . I have shaded this area in the first diagram.

    Therefore the value of the integral = Area of the shaded region.

    We will find this area using geometry. We need to find area ODAC.

    Area ODAC =Area OBC - Area of OAB + Area of OAD(this is a ...

    Solution Summary

    Very detailed solutions are provided in a 3-page word document along with 2 diagrams and 3 charts.