Dabke Dance and Mansaf Dish: Cultural Traditions of the Levant
"Discover the vibrant cultural traditions of the Levant region with the joyful dabke dance, a traditional form of line and circle dancing originating in the Middle East. Learn about where and when dabke is performed, who practices it, and the significance it holds in sharing historical narratives. Explore the delicious Jordanian dish of mansaf, made with lamb in fermented dried yogurt sauce. Immerse yourself in the rich heritage of these practices that bring people together in celebrations and gatherings."
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Presentation Transcript
In this sequence we review and extend the use of definite integrals to represent plane areas. Recall that the integral ?? ? ?? ? measures the area between the graph of ? and the ? axis from ? = ? to ? = ?, but treats as negative any part of this area that lies below the ? axis. (We are assuming that ? < ?.)
In order to express the total area bounded by ? = ?(?),? = ? ,? = ?, and ? = ?, counting all of the area positively, we should integrate the absolute value of ?(see the figure below): ?? ? ?? = ?? ?? and ??(?) ?? = ??+ ??. ? ?
There is no rule for integrating ??(?) ?? ? one must break the integral into a sum of integrals over intervals where ? ? > ? so ?(?) = ?(?), and intervals where ? ? < ? so ?(?) = ?(?). To find the area between the graph of ? and the ? axis over the interval [?,?]: 1. Subdivide [?,?] at the zeros of ?. 2. Integrate ? over each subinterval. 3. Add the absolute values of the integrals.
Example: The figure below shows the graph of the function ? ? = sin? between ? = ? and ? = ??. Compute the area between the graph of and the ? axis over [?,??]. Solution: First find the zeros of ?. On [?,??], the zeros of ? ? = sin? are ? = ? ,? ,??.
So, the area between the graph of ?(?) and the ? axis over [?,??] is calculated by breaking up the domain of sin? into two pieces: the interval [?,?] over which it is nonnegative and the interval [?,??] over which it is non positive. ? sin??? = ] cos?? ?= cos? cos? = ? ? ?? ??= cos2? cos? = ? ] sin??? = cos?? ? The second integral gives a negative value. The area between the graph and the axis is obtained by adding the absolute values Area ? + ? = ?
Example: Find the area of the region between the ? axis and the graph of ? ? = ?? ?? ?? over [ ?,?]. Solution: First find the zeros of ?. Since ? ? = ?? ?? ?? = ? ?? ? ? = ?( ) ? ? + ? the zeros are ? = ? , ? and ? (see the figure below). ?
The zeros subdivide [?,?] into two subintervals:[?,?] on which ?(?) ? and [?,?], on which ?(?) ?. We integrate ? over each subinterval and add the absolute values of the calculated integrals. ? ?? ?? ?? ?? = ? ?? ? ?? ? ?? ? ?? = ? ? ? ? ?? ? ?? = ? ?? ?? ?? ?? = ? ?? ? ? ? The total enclosed area is obtained by adding the absolute values of the calculated integrals. ? ??+ ? ?=?? ???? = ??
Areas Between Curves: Suppose we want to find the area of a region that is bounded above by the curve ? = ?(?), below by the curve ? = ?(?) and on the left and right by the lines ? = ? and ? = ? (see figure below). The region might accidentally have a shape whose area we could find with geometry, but if ? and ? are arbitrary continuous functions, we usually have to find the area with an integral.
Definition: If ? and ? are continuous throughout [?,?], then the area of the region between the curves ?(?) and ?(?) from ? to ? is the integral of ? from ? to ?: ?? ? ?(?) ?? When applying this definition it is helpful to graph the curves. It helps you find the limits of integration if they are not given. You may need to find where the curves intersect to determine the limits of integration, and this may involve solving the equation ? ? = ?(?) for values of ?. ? ? = ?
Then you can integrate the function for the area between the intersections. Example: Find the area of the region bounded above by the curve ? = ?? ?+ ?, below by the curve ? =? right by ? = ?. Solution: The figure in the next slide displays the graphs of the curves and the region whose area we want to find. ???, on the left by ? = ?, and on the
The area between the curves over the interval is given by ? ?? ?+ ? ? ???? = = ? ? ? ?.???? ? ?? ?+?? ? ? ??? ? ? ? ?
Example: Find the area of the region bounded by the curves ? = ????,? = ???? ,? = ? , and ? = ?/? Solution: The points of intersection occur when ???? = ????, that is, when ? = ?/? since ? ? ?/?. The intersection point ? = ?/? subdivide [?,?/?] into two subintervals: [?,? on which ???? ????. ?] on which ???? ???? and [? ?,? ?] ,
We integrate sin? cos? over each subinterval and add the absolute values of the calculated integrals. ?/? (sin? cos?) = cos? sin?? ?/?= ? ? ? ?/? ?/?= ? + ? (sin? cos?) = cos? sin??/? ?/? Therefore the required area is: ? ? + ? + ? = ? ? ?
