Calculus Calculus answers two very important questions The first how to find the instantaneous rate of change we answered with our study of derivatives The second we are now ready to answer how to find the area of irregular regions ID: 552121
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Slide1
2. Definite Integrals and Numeric IntegrationSlide2
Calculus
Calculus answers two very important questions.
The first, how to find the instantaneous rate of change, we answered with our study of derivatives
The second we are now ready to answer, how to find the area of irregular regions.Slide3
Approximating Area
We will now approximate an irregular area bounded by a function, the x-axis between vertical lines x=a and x=b, like the one below by finding the areas of many rectangles and summing them up.Slide4
Finding area
Break region into subintervals (strips)
These strips resemble rectangles
Sum of all the areas of these “rectangles” will give the total area Slide5
Rectangular Approximation Method (RAM)
Since the height of the rectangle varies along the subinterval, in order to find area of the rectangle, we must use either the left hand endpoint (LRAM) to find the height, the right hand endpoint (RRAM) or the midpoint (MRAM)
The more rectangles you make, the better the approximationSlide6
Rectangular Approximation Method (RAM)
If a function is increasing, LRAM will underestimate the area and RRAM will overestimate it.
If a function is decreasing, LRAM will overestimate the area and RRAM will underestimate itSlide7
Trapezoid Approximation
Another approximation we can use (and probably the best) is trapezoids.
Trapezoids give an answer between the LRAM and RRAM
The formula for the area of a trapezoid is ½(x)(y
1
+y
2
)Slide8
Example 1
Find the area under y=x
2
+2x-3 from x=0 to x=2, use width of ½
LRAM
MRAM
RRAM
TrapezoidSlide9
Example 2
We can also
approximate integrals when
our function is given to us in either data form.Approximate using LRAM, MRAM, RRAM, and trapezoids.
Also
approximate f’(1)Slide10
Example 3
Approximate using LRAM, RRAM, and trapezoids.
Why
did I leave off MRAM? Also approximate f’(7)Slide11
How many rectangles should we make?
The estimate of area gets more and more accurate as the number of rectangles (n) gets largerSlide12
How many rectangles should we make?
If we take the limit as n approaches infinity, we should get the exact area
We will talk more about this tomorrow…..Slide13
BREAK!!Slide14
Remember from yesterday…..
We were talking about increasing the number of rectangles giving us a better estimate of the area
What if we took the limit as n approached infinity??
The area approximation would approach the actual areaThe process of finding the sum of areas of rectangles to approximate area of a region is called a Riemann Sum, after Bernhard RiemannSlide15
Riemann Sums
Riemann proved that the finite process of adding up the rectangular areas could be found by a process known as definite integration. Here is the essence of his great, time-saving work.Slide16
Example 4
Evaluate geometrically as well as on your calculatorSlide17
Negative area?
The example we just looked at was non-negative on the interval we evaluated. This is not always the case.
If f(x) is non-negative and
integratable over a closed interval [a,b
] then the area under the curve is the definite integral of f from a to b
If f(x) is negative and integrable over a closed interval [
a,b
], then the area under the curve is the OPPOSITE of the definite integral of f from a to b.
Slide18
In general…
does NOT give us area but rather the NET accumulation over the interval x=a to x=b. If f(x) is positive and negative on a closed interval, then will NOT give us area.Slide19
When integrating left to right, regions above the x-axis are positive and regions below the x-axis are negative.
When integrating right to left, regions above the x-axis are negative and regions below the x-axis are positive.
This can be summarized as Slide20
Negative functions
When using definite integrals to find area, you must divide the interval into subintervals
where function is positive and where it is
negative and use absolute values of definite integralWhen using area to find definite integrals, you must assign the correct sign to the area.Slide21
Example 5
The graph of f(x) is shown below. If A
1
and A2 are positive numbers that represent the areas of the shaded regions, then find the following.Slide22
Another property of definite integrals
The property that allows us to do the calculations in the previous example isSlide23
Example 6
Approximate using four subintervals of equal length and trapezoidal method. Test your answer against the calculator’s approximation using
fnint
. Can any of these approximations represent the area of the region? Why or why not? Slide24
Example 7
Find the area in the previous problem using trapezoids and also set up integrals needed to calculate with calculator.Slide25
Another way to find area with calculatorSlide26
Example 8
Sketch the region corresponding to each definite integral, then evaluate each integral using a geometric formula. Decide if the integral represents the area.Slide27
Example 9
Evaluate and . Do these represent the area of the region? Why or why not? If not, what is the area of the region?Slide28
Properties of definite integrals
We have seen some of these already.Slide29
Example 10
Given that
FindSlide30
Example 11
If and , find