Final answer:
The area between the curves x=sin(5y) and x=1-cos(5y) over the interval -π/10 to π/10 involves setting up and evaluating an integral of the difference between the two functions.
Step-by-step explanation:
To find the area between two curves, we typically integrate the difference between the top function and the bottom function over the given interval. In this case, the curves are defined by x=sin(5y) and x=1-cos(5y), and we are interested in the interval -π/10 ≤ y ≤ π/10.
Since sin(5y) and cos(5y) differ by a phase shift of π/2, their squares average to the same value over one period. This suggests that we can use trigonometric identities to simplify our calculation. Specifically, we use the identity cos2(θ) = 1 - sin2(θ) to express one function in terms of the other and integrate accordingly.
The exact area can be found by setting up the integral:
\int_{-\pi/10}^{\pi/10}[1-cos(5y) - sin(5y)]dy
Since the functions are periodic and symmetric, and the interval corresponds to a full period, we can expect the areas under each half of the curve to cancel out to some extent, simplifying the calculation.