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apples
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The problem is related to polar curves. most of the topics i need to do are easy (finding the slope, finding the area etc.)
What I'm facing problems with is that when I find the area, I don't know how to find the limits.
Sample problem:
Find the area of the region in the plane enclosed by the cardioid r=4+(4sinθ)
b
A = ∫ (1/2)(r^2) dθ
a
Graphing the curve is no biggie, I use my calculator. The problem is when I use the equation to find the Area, I don't know what the interval is I don't know 'b' and 'a' are.
I have a calculus book, an AP calculus book, and a pre-calculus book. None of the books tell you how to find the interval. In their solved questions, they tell you that in the solution. The problem I wrote above is an example also.
They tell you the interval is (for the q above) 0 to 2π(pi)
but how do i know that what about other questions.
Another example:
Find the area inside the smaller loop of the limacon r=1+(2cosθ)
Here they say the limits are (2π(pi))/(3) to (4π(pi))/3
The exact explanation to this in the book is:
(for first question) "Because r swoops out the region as θ goes from 0 to 2π(pi), these are our limits of integration.
(For 2nd q) Because in the inner loop, r sweeps out the region as θ goes from (2π(pi))/(3) to (4π(pi))/3, these are our limits of integration.
But why!? What does r swoops mean. How do I know the limits!?
please help!
What I'm facing problems with is that when I find the area, I don't know how to find the limits.
Homework Statement
Sample problem:
Find the area of the region in the plane enclosed by the cardioid r=4+(4sinθ)
Homework Equations
b
A = ∫ (1/2)(r^2) dθ
a
The Attempt at a Solution
Graphing the curve is no biggie, I use my calculator. The problem is when I use the equation to find the Area, I don't know what the interval is I don't know 'b' and 'a' are.
I have a calculus book, an AP calculus book, and a pre-calculus book. None of the books tell you how to find the interval. In their solved questions, they tell you that in the solution. The problem I wrote above is an example also.
They tell you the interval is (for the q above) 0 to 2π(pi)
but how do i know that what about other questions.
Another example:
Find the area inside the smaller loop of the limacon r=1+(2cosθ)
Here they say the limits are (2π(pi))/(3) to (4π(pi))/3
The exact explanation to this in the book is:
(for first question) "Because r swoops out the region as θ goes from 0 to 2π(pi), these are our limits of integration.
(For 2nd q) Because in the inner loop, r sweeps out the region as θ goes from (2π(pi))/(3) to (4π(pi))/3, these are our limits of integration.
But why!? What does r swoops mean. How do I know the limits!?
please help!