Finding the area of a disk divided by a parabola function.

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SUMMARY

The discussion centers on finding the area of a disk defined by the equation x² + y² ≤ 8, which is divided by the parabola y = 1/2 x². Participants express confusion regarding the problem's validity, noting that the two areas created by the intersection of the curves are not equal. The need for clarification on the intersection points and the method to calculate the area under a curve is emphasized, indicating a potential transcription error in the problem statement.

PREREQUISITES
  • Understanding of calculus concepts, particularly integration.
  • Familiarity with the equations of curves, specifically parabolas and circles.
  • Knowledge of finding intersection points between functions.
  • Ability to visualize geometric shapes and areas in the Cartesian plane.
NEXT STEPS
  • Learn how to calculate the area under a curve using definite integrals.
  • Study methods for finding intersection points between two functions.
  • Explore the application of integration in calculating areas between curves.
  • Review the properties of parabolas and circles in coordinate geometry.
USEFUL FOR

Students in calculus courses, mathematics educators, and anyone interested in solving geometric area problems involving curves and integration.

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Homework Statement


The parabola y=1/2 x^2 divides the disk x^2+y^2 <or= to 8 into two equal parts. Find the area of both parts.


Homework Equations





The Attempt at a Solution


I have no idea of how to go about solving this. We haven't done any application problems until now and when I transferred colleges they were several chapters ahead so I'm struggling as it is. If someone could give me a system to go about solving these and point me in the right direction I'd be very grateful.
 
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Do you know how to calculate an area under a curve?
Do you know how to calculate an area between two functions?

Those two parts don't look equal to me.
 
Drawing a picture seems like a good start. In particular, at what points do the two curves intersect?
 
mfb said:
Those two parts don't look equal to me.
Agree, this question seems wrong as stated. One of the two "halves" will contain the entire bottom half of the disk and then some.
 
I suspect a transcription error. Maybe the original says "unequal".
 
Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?

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