2 nd problem of calculus – area

2nd Problem of Calculus – Area •Break curve into rectangles•Total area: add

sigma summation

•Underestimate or overestimate?

𝑎1𝑎2𝑎3𝑎4

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 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

endpointsheight =

right sum of 4 areas

right sum = •Width = •Height =

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

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

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

x 0 2 4 6 8 10 12

f(x)

3 7 19 39 67 103

147

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

Most people don't know that polar bears 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.

Trapezoid Example

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:

Location of measurement

Left endpt

2nd value

3rd value

4th value

5th value

6th value

7th value

8th value

Right endpt

Measurement

14 16 20 24 22 18 16 14 12

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 unitsArea of trapezoid = ½ base (height1+ height2)Area

square bears

Location of measurement

Left endpt

2nd value

3rd value

4th value

5th value

6th value

7th value

8th value

Right endpt

Measurement(bear units)

14 16 20 24 22 18 16 14 12

4[ (14+16 )+ (16+20 )+ (20+24 )+(24+22 )+¿ (22+18 )+ (18+16 )+ (16+14 )+(14+12)]

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

Review – Approximating AreasRectangles 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.