Integral Question: Areas between curves?

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SUMMARY

The discussion focuses on calculating the area between the curves defined by the equations y=4cos(x) and y=6sin(2x) in the first quadrant, specifically for x values between 0 and pi/2. The intersections of the curves occur at x=0 and x=arcsin(1/3). A participant repeatedly arrives at an incorrect area calculation of 10/3, indicating a misunderstanding of the integration limits or the region being analyzed. The correct approach requires careful sketching of the area and proper identification of the relevant intersection points.

PREREQUISITES
  • Understanding of integral calculus, specifically area between curves.
  • Familiarity with trigonometric functions and their graphs.
  • Knowledge of integration techniques with respect to x and y.
  • Ability to find intersections of curves algebraically.
NEXT STEPS
  • Review the method for finding the area between curves using definite integrals.
  • Study the properties and graphs of the functions y=4cos(x) and y=6sin(2x).
  • Learn how to determine the correct limits of integration based on intersection points.
  • Practice sketching regions defined by multiple curves to visualize integration areas.
USEFUL FOR

Students studying calculus, particularly those focusing on integral applications, as well as educators looking for examples of area calculations between curves.

baquid
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Homework Statement



Sketch the region in the first quadrant enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region.
y=4cosx, y=6sin2x, x=0.

Homework Equations





The Attempt at a Solution



I got the intersections, which are x=pi/2, arcsin1/3 as well as x=0 but I seem to keep getting 10/3, which is not the right answer.
 
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baquid said:

Homework Statement



Sketch the region in the first quadrant enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region.
y=4cosx, y=6sin2x, x=0.

Homework Equations





The Attempt at a Solution



I got the intersections, which are x=pi/2, arcsin1/3 as well as x=0 but I seem to keep getting 10/3, which is not the right answer.

You should really show more of your work if you expect people to understand what region you are integrating over, but I think you are integrating over the wrong region. x=0 is the y-axis, and I don't see how the intersection at x=pi/2 is relevant.
 

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