Area under a curve

1. Area under a curve 5.1 -5.4 Day 2

2. Approximating Area of a plane region Inscribed Rectangles

3. Circumscribed Rectangles

4. Finding Area by a limit Definition Starting interval

6. Finding Area by limit Def. Cont.

7. Generalize the procedure for Area • Y=f(x) • Region is bounded by x axis and vertical asymptotes of x=a and x=b • [a, b] – subdivide this interval into n subintervals

8. Def of Area of Region in the plane • Let f be continuous and non-negative on the interval [a, b]. The area of the region bounded by the graph of f, the x-axis and the vertical lines x=a and x=b is

9. Def of Riemann Sum • Let f be defined on the closed interval [a, b] and let ∆ be an arbitrary partition of [a, b], where ∆xi is the width of the ith subinterval. If ci is any point in the ith subinterval, then the sum Is called a Riemann sum of f for the partition of ∆

10. Riemann Cont • Norm of the Partition- width of the largest subinterval • Regular Partition – every subinterval is of equal width • General Partition -

11. Def of the Definite Integral • If f is defined on the closed interval [a, b] and the limit of a Riemann Sum of f exists, then we say f is integrable on [a, b] denoted by • The limit is called the definite integral of f from a to b. The number a is the lower limit of integration and the number b is the upper limit of integration. • Definite Integral number • Indefinite Integral family of functions