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MAT 1235 Calculus II Section 5.1 Area Between Curves http://myhome.spu.edu/lauw
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Homework WebAssign HW 5.1
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Preview Want to compute the area of the region between two curves. Two types of basic regions Bounded above and below Bounded on the left and on the right Composite region Break the region up into smaller basic regions
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The Area Problem
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Two Important Themes The formula for computations The process of finding the formula More important in a long run The processes of finding integral based formula (geometric properties, physical properties) are similar. Volume, Surface area, Arc length, Work Done, Center of Mass, …
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Recall: Definition
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In other words… Any limit of the form : can be interpreted as
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Type I Region: Region Bounded Above/Below
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i th subinterval sample point
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Type I Region: Region Bounded Above/Below Area of each rectangle: Area of the region:
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Type I Region: Region Bounded Above/Below Area of each rectangle: Area of the region: Key: To know which curve is above/below
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Example 1 Find the area of the region enclosed by and
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Example 1 Step 1: Sketch the graphs
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Type II Region: Region Bounded Left/Right
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Example 2 Find the area of the region enclosed by and
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Example 2 >with(plots): >implicitplot([x=y^2-1,x=y+1],x=-2..4,y=-2..4);
![Example 2 >with(plots): >implicitplot([x=y^2-1,x=y+1],x=-2..4,y=-2..4);](http://images.slideplayer.com/27/9174597/slides/slide_17.jpg "Example 2 >with(plots): >implicitplot([x=y^2-1,x=y+1],x=-2..4,y=-2..4);")
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Example 2
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Summary 1. Sketch the curves Determine the type of the region (above/below, left/right) Determine the nature of each curve 2. Find the upper/lower limits 3. Compute the integrals
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