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Chapter 9: Q. 3. (page 755)

When we use rectangular coordinates to approximate the area of a region, we subdivide the region into vertical strips and use a sum of areas of rectangles to approximate the area. Explain why we use a 鈥渨edge鈥� (i.e., a sector of a circle) and not a rectangle when we use polar coordinates to compute an area.

Short Answer

Expert verified

The region's area is approximated using "wedges," which refers to a circle's sector.

A wedge-shaped slice of the region is determined by a minor change in螖胃

Step by step solution

01

Given information

Consider a function r=f(胃)in polar coordinates. Where 胃is angular rotation

02

Explain why we use a “wedge” and not a rectangle when we use polar coordinates to compute an area.

A circle's area is calculated by splitting it into an infinite number of wedges produced by radii drawn from the center. When these wedges are rearranged, they form a rectangle with a height equal to the circle's radius and a base length equal to half of the circle's diameter.

The region's area is approximated using "wedges," which refers to a circle's sector.

A wedge-shaped slice of the region is determined by a minor change in螖胃

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Most popular questions from this chapter

In Exercises 32鈥�47 convert the equations given in polar coordinates to rectangular coordinates.

r2=cos胃

In Exercises 24鈥�31 find all polar coordinate representations for the point given in rectangular coordinates.

0,0

Use Cartesian coordinates to express the equations for the ellipses determined by the conditions specified in Exercises 32鈥�37.

foci0,卤332, minor axis3

Use Cartesian coordinates to express the equations for the ellipses determined by the conditions specified in Exercises 32鈥�37.

foci(卤1,0),major axis4

Let A>B>0. Show that the distance from any point on the graph of the curve with equation x2A2+y2B2=1to the point (-C,0)isD2=A2+CxA2, whereC=A2-B2.
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