Table of Contents 29. Section 5.1 Approximating and Computing Area.

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Presentation transcript:

Table of Contents 29. Section 5.1 Approximating and Computing Area

Approximating and Computing Area Essential Question – What is the other major topic of calculus?

3 main concepts Limits Derivatives Integrals

Integrals Describe how the instantaneous changes of functions (derivatives) can accumulate over an interval to produce the function

Finding area Break region into subintervals (strips) These strips resemble rectangles Sum of all the areas of these “rectangles” will give the total area

Example A particle starts at x=0 and moves along the x-axis with velocity v(t)=t 2. Where is the particle at t=3? Each interval is ¼ width. Find height at each midpoint of interval. Multiply height times width to get area. Sum all the areas. Add all 1/256+9/256+25/256+49/ / / / / / / / /256 = 8.98 Subinterval[0, ¼][¼, ½][½,3/4][3/4, 1]etc Midpoint1/83/85/87/8 Height (t 2 )1/649/6425/6449/64 Area1/2569/25625/25649/256

Rectangular Approximation Method (RAM) In the last example, we used the midpoint RAM (MRAM) (because we used the midpoint of the interval) We can also use the left hand endpoint (LRAM) Or the right hand endpoint (RRAM) If the function is monotonic (either increasing or decreasing), the actual area lies somewhere between LRAM and RRAM

Example Find the area under y=x 2 from x=0 to x=3, use width of ½ LRAM MRAM RRAM If the subinterval gets smaller and smaller, these will all equal each other

Example Find the area under y=x 2 +2x-3 from x=0 to x=2, use width of ½ LRAM MRAM RRAM

Cardiac Output The number of liters your heart pumps in a fixed time interval is called cardiac output. At rest, it is about 5-6 L/min. With strenuous exercise, it can be as high as 30 L/min. Dye is injected near the heart and its concentration is measured to find cardiac output.

Example Each rectangle has a width of 2 units. Use MRAM to find area and cardiac output. If 5.6 mg was injected,

Summation Notation means sum K tells where to start and end summing

Writing RAM in summation notation

How many rectangles should we make? The estimate of area gets more and more accurate as the number of rectangles (n) gets larger If we take the limit as n approaches infinity, we should get the exact area We will take more about this tomorrow…..

Book notation If interval is 3 units long and you have 6 subintervals, Each subinterval will be 3/6 or ½ wide.

Assignment Pg. 308: #1-25 odd