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Ch 8.3 Wkst AP Calculus BC Name: Area, Arc Length, and Slope of Polar Coordinates

Objective: to use integrals to calculate area, arc length, and slope of Polar Coordinates.

For a polar curve, [pic], with [pic], the Area from [pic] to [pic] is [pic].

For a polar curve, [pic], the Arc Length of the Polar Curve = Distance Traveled from [pic] to [pic] ,

[pic], where .

This can be shown to be equivalent to

[pic] . .

For a polar curve, [pic], the Slope of the Polar Curve at [pic] is [pic] , where .

Example 1: Find the area inside the cardioid [pic] for [pic]. NO CALCULATOR.

Example 2: Find the length of the polar equation [pic] for [pic]. CALCULATOR REQUIRED.

Example 3: Find the slope of the curve [pic] at [pic]. NO CALCULATOR.

Ch 8.3 Wkst AP Calculus BC Name: Area, Arc Length, and Slope of Polar Coordinates

For a polar curve, [pic], with [pic], the Area from [pic] to [pic] is [pic].

For a polar curve, [pic], the Arc Length of the Polar Curve = Distance Traveled from [pic] to [pic] ,

[pic], where .

This can be shown to be equivalent to

[pic] . .

For a polar curve, [pic], the Slope of the Polar Curve at [pic] is [pic] , where .

1. Find the area bounded by the spiral [pic] and [pic]. NO CALCULATOR.

2. Find the area of one petal of the rose [pic]. NO CALCULATOR.

3. Find the area of the region that is bounded by the graphs of the [pic], [pic], and [pic].

CALCULATOR REQUIRED.

4. Find the area inside the cardioid [pic] and outside the circle [pic]. CALCULATOR REQUIRED.

5. Find the length of the spiral [pic] for [pic]. CALCULATOR REQUIRED.

6. Find the length of one petal of the rose [pic] for [pic]. CALCULATOR REQUIRED.

7. If [pic], then at what values of [pic] is [pic] on [ 0 , 2π ]?

CALCULATOR REQUIRED.

8. For what values of [pic] on the polar curve [pic], for [pic], are the tangent lines horizontal? vertical?

CALCULATOR REQUIRED.

9. Find the maximum value of the y-coordinate of points on the polar curve [pic].

SCIENTIFIC CALCULATOR REQUIRED, BUT NO GRAPHING CALCULATOR.

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[pic]

[pic]

ANSWERS:

1) [pic]

2) h–XØh²2œCJEHöÿU[pic]j/€?Z[pic]h²2œU[pic]V[pic]hÓX:h²2œCJaJ

h²2œCJaJ

h²2œCJaJh²2œ>*[pic]CJaJh²2œCJmHsHh²2œ>*[pic]CJaJhY7qh²2œ>*[pic]CJ

h²2œCJaJh–XØh EMBED Equation.3 [pic]

3) 1.246

4) 4.134

5) 21.256

6) 4.712

7) 1.834, 2.699, π, 5.678

8) Horz: 0, 2.029

Vert: 0.860

9) 1.760

[pic]

[pic]

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