MAT 1235 Calculus II Section 5.1 Area Between Curves

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Presentation transcript:

MAT 1235 Calculus II Section 5.1 Area Between Curves

Homework WebAssign HW 5.1

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

The Area Problem

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, …

Recall: Definition

In other words… Any limit of the form : can be interpreted as

Type I Region: Region Bounded Above/Below

i th subinterval sample point

Type I Region: Region Bounded Above/Below Area of each rectangle: Area of the region:

Type I Region: Region Bounded Above/Below Area of each rectangle: Area of the region: Key: To know which curve is above/below

Example 1 Find the area of the region enclosed by and

Example 1 Step 1: Sketch the graphs

Type II Region: Region Bounded Left/Right

Example 2 Find the area of the region enclosed by and

Example 2 >with(plots): >implicitplot([x=y^2-1,x=y+1],x=-2..4,y=-2..4);

Example 2

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