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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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