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2 nd Problem of Calculus – Area

2 nd Problem of Calculus – Area . Break curve into rectangles Total area: add sigma  summation Underestimate or overestimate?. Sigma Notation Examples. Summation (Sigma) Formulas. Example. Homework: p. 262 #1, 5, 12, 16, 21, 22, 29, 31. Find Area Between a Curve and X-axis.

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2 nd Problem of Calculus – Area

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  1. 2nd Problem of Calculus – Area • Break curve into rectangles • Total area: add sigma  summation • Underestimate or overestimate?

  2. Sigma Notation Examples

  3. Summation (Sigma) Formulas

  4. Example Homework: p. 262 #1, 5, 12, 16, 21, 22, 29, 31

  5. Find Area Between a Curve and X-axis on [0, 1] using right-hand rectangles • Overestimate • Area of rectangle = length x width (width x height) • Width of each rectangle = • Height of each = function value at right endpoints height = right sum of 4 areas

  6. right sum = • Width = • Height =

  7. How can we get a closer approximation? use more rectangles (exact area) *As n gets large, any term with n on the bottom gets smaller  approaches zero

  8. Approximate the area under on [0, 4], n = 10, left endpoints • For left endpoints, use instead of • Width = • Height =

  9. Let f be a continuous function defined on [0, 12] as shown below. Find the midpoint Riemann sum for f(x) over [0, 12] using 3 subintervals of equal length. Width = Height = f(x) at midpoints

  10. Find the exact area using right endpoints. on [0, 2] • Width = • Height = Exact area:

  11. Trapezoid Example Most people don't know that polarbears live in igloos they build each year. To build an igloo,they find a large field of snow and flatten it down so the snow is compressed sufficiently. Next, they cut blocks of snow from this field and build igloos by layering the block in smaller and smaller rings until they have formed their structure. It takes approximately 100 blocks of snow to form one igloo.

  12. This year they came upon a field which they had to measure using calculus. Blocks are measured in "cubic bears" and length is measured in "bears". Basically, a block of snow to make an igloo is 1/8 of a bear wide and 1/4 of a bear long. The height of the block corresponds to the depth at which they cut into the snow and therefore does not impact this problem. The field of snow they used this year was divided up into subintervals of 4 bear units. The length of the field was measured at each interval and the following distances were calculated:

  13. Using the trapezoidal rule, calculate the area of the field (indicate correct units). Show the work used to find this solution. Width of each subinterval = 4 bear units Area of trapezoid = ½ base (height1+ height2) Area square bears

  14. How many blocks of snow can be made? • 1 block of snow = square bears • Total # blocks = blocks How many igloos can be built? • 100 blocks for each igloo • igloos with 4 blocks left

  15. Review – Approximating Areas Rectangles with equal subintervals: • Width = • Height = function value • Area = Trapezoids: Area = If you are given a table, do not use ’s. just means “sum.” Add up all of your pieces without a function.

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