1. 4.2 – Areas 4.3 – Riemann Sums
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2. The Area Problem
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3. Area Problem
Our idea, as with derivatives earlier, is to use a limit of approximations to arrive at the true area.
We…
first approximate the region S by rectangles and then
take the limit of the areas of these rectangles as we increase the number of rectangles.
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4. Example 1
Use rectangles to estimate the area under the parabola y = x2 from 0 to 1.
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5. Example 1 (solution)
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6. Example 1 (solution)
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7. Definition
We define the area A to be the limit of the sums of the areas of the approximating rectangles:
The width of each of the n strips is
These strips divide the interval [a, b] into n subintervals
where x0 = a and xn = b.
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8. Using Right Endpoints The right endpoints of the subintervals are:
Then the area of the ith rectangle is f(xi)∆x.
So the area of S is approximated by the sum of the areas of these rectangles:
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9. More Generally
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10. More Generally (cont’d)
Since this approximation seems to become better and better as n increases, we define the area A of the region S as follows:
Sigma Notation:
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11. Example 2
Let A be the area of the region that lies under the graph of f(x) = e–x between x = 0 and x = 2.
(a) Using right endpoints, find an expression for A as a limit. Do not evaluate the limit.
(b) Estimate the area by taking the sample points to be midpoints and using
four subintervals
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12. distance = velocity * time
The Distance Problem
Here our goal is to find the distance traveled by an object during a certain time period if the velocity is known at all times.
If the velocity is constant, then the problem reduces to the familiar formula
distance = velocity * time
But what if the velocity varies?
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13. Example 3
Suppose the odometer on our car is broken and we want to estimate the distance driven over a 30-second time interval.
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14. Connection With Area On the next slide we
sketch the velocity function of the car, and
draw rectangles whose heights are the initial velocities for each time interval.
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15. Connection With Area
Thus, the distance traveled is equal to the area under the graph of the velocity function.
So the distance problem and the area problem are the same problem.
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