Find Area Under x=2Sin^2(y) & y=x^2 Graphs

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In summary, to find the area of the regions shown in the figures, one must set the two equations y = x^2 and x = 2 Sin ^2 (y) equal to each other to find the points of intersection. However, there may be difficulty in simplifying for y, so one can use the identity cos(2y)=1-2sin^{2}(y) to rearrange and find y as a function of x.
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FuturEngineer
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Find the area of the regions shown in the figures.

These are the graphs used :

y = x^2
x = 2 Sin ^2 (y)

I know that I need to set the two equations equal to each other in order to find the points of intersection, but I run into some trouble when trying to simplify it for y.

This is what I tried

Since y=x^2
Sqrt.(y)= x

So now I have
x= Sqrt.(y)
x= 2 Sin ^2 (y)

Sqrt.(y) - 2 Sin ^2 (y) = 0

So based on the graph that I am given in the book I can't really tell which is the top or right curve to subtract the bottom or left curve.

After this step I would square both sides and be left with (Sqrt.(y) -2 Sin(^2y)^2) = 0, but I am not sure if that is allowed. Not sure on how to proceed from here. I would appreciate some help. Thanks!
 
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Hint: [itex] cos(2y)=1-2sin^{2}(y)[/itex]. Rearrange and find y as a function of x.
 

Related to Find Area Under x=2Sin^2(y) & y=x^2 Graphs

What is the equation for the graph of x=2Sin^2(y) & y=x^2?

The equation for this graph is x=2Sin^2(y) & y=x^2. This means that the x-coordinate is equal to 2 times the sine squared of the y-coordinate, and the y-coordinate is equal to the x-coordinate squared.

How do you find the area under the graph of x=2Sin^2(y) & y=x^2?

To find the area under this graph, you can use the formula for calculating the area under a curve: A = ∫f(x)dx. Since the graph is defined by two equations, you will need to use the double integral: A = ∬f(x,y)dxdy. You can then integrate over the appropriate bounds to find the area.

What are the bounds for finding the area under x=2Sin^2(y) & y=x^2?

The bounds for finding the area under this graph will depend on the limits of integration for both the x and y coordinates. You will need to determine the maximum and minimum values for both x and y in order to set the appropriate bounds for the double integral.

How can you graph the equation x=2Sin^2(y) & y=x^2?

To graph this equation, you can use a graphing calculator or software such as Desmos or GeoGebra. Simply input the equations and the software will plot the graph for you. Alternatively, you can also plot the points by hand by choosing values for x and solving for y, then plotting the points on a coordinate plane.

What is the relationship between the x and y coordinates on the graph of x=2Sin^2(y) & y=x^2?

The relationship between the x and y coordinates on this graph is that they are defined by two equations, x=2Sin^2(y) and y=x^2. This means that the x-coordinate is dependent on the value of the y-coordinate, and vice versa. This can be seen in the shape of the graph, where the curve of x=2Sin^2(y) changes as the y-coordinate changes due to the equation y=x^2.

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