# Revision history [back]

### How to use sage to solve this problem?

Suppose we have an equation: $${(y\cos{(x)}+u})^{2}+{(-(y\sin{(x)}+v)^3+1)}^{\frac{2}{3}}=1$$

The graph of this implicit relation has two regions, where one part is above the x-axis, and the other part is below the x-axis.

Suppose we're are trying to find the area of the relation above the x-axis, from the domain ${0}\le{x}\le{2\pi}$. How can one solve for the values of u and v that can give the highest area for this equation that's is above the x-axis.

Here is a brief illustration:

Copy and paste the link into the internet:

If you find the picture try and use sage to graph the equation above, with respect to u and v.

### How to use sage to solve this problem?

Suppose we have an equation: $${(y\cos{(x)}+u})^{2}+{(-(y\sin{(x)}+v)^3+1)}^{\frac{2}{3}}=1$$

The graph of this implicit relation has two regions, where one part is above the x-axis, and the other part is below the x-axis.

Suppose we're are trying to find the area of the relation above the x-axis, from the domain ${0}\le{x}\le{2\pi}$. How can one solve for the values of u and v that can give the highest area for this equation that's is above the x-axis.

Here is a brief illustration:

Copy and paste the link into the internet: If you find the picture try Note: The blue stripes should stop at $2\pi$, and use sage to graph if the equation above, with respect has regions that "UNDEFINED", try to u and v.ignore it.

### How to use sage to solve this problem?

Suppose we have an equation: $${(y\cos{(x)}+u})^{2}+{(-(y\sin{(x)}+v)^3+1)}^{\frac{2}{3}}=1$$

The graph of this implicit relation has two regions, where one part is above the x-axis, and the other part is below the x-axis.

Suppose we're are trying to find the area of the relation above the x-axis, from the domain ${0}\le{x}\le{2\pi}$. How can one solve for the values of u and v that can give the highest area for this equation that's is above the x-axis.

Here is a brief illustration: Note: The blue stripes should stop at $2\pi$, and if the equation has regions that are "UNDEFINED", try to ignore it.

### How to use sage to solve this problem?

Suppose we have an equation: $${(y\cos{(x)}+u})^{2}+{(-(y\sin{(x)}+v)^3+1)}^{\frac{2}{3}}=1$$

The graph of this implicit relation has two regions, where one part is above the x-axis, and the other part is below the x-axis.

Suppose we're are trying to find the area of the relation above the x-axis, from the domain ${0}\le{x}\le{2\pi}$. How can one solve for the values of u and v that can give the highest area for this equation that's is above the x-axis.x-axis, and is still continuous upper half relation above the x-axis

Here is a brief illustration: Note: The blue stripes should stop at $2\pi$, and if the equation has regions that are "UNDEFINED", try to ignore it.

### How to use sage to solve this problem?Maximize the integral of an implicit function with two parameters

Suppose we have an equation: $${(y\cos{(x)}+u})^{2}+{(-(y\sin{(x)}+v)^3+1)}^{\frac{2}{3}}=1$$$${(r\cos{(t)}+u})^{2}+{(-(r\sin{(t)}+v)^3+1)}^{\frac{2}{3}}=1$$

The graph of this implicit relation has two regions, where one part is above the x-axis, and the other part is below the x-axis.

Suppose we're are trying to find the area of the relation above the x-axis, from between the domain x-values ${0}\le{x}\le{2\pi}$. How can one solve for the values of u and v that can give the highest area for this equation that's is above the x-axis, so that u and v as point (u,v) is still continuous upper half inside the implicit relation above the x-axis

Here is a brief illustration: $$(1-x^2)^3=(1-y^3)$$.

Note: The blue stripes should stop at $2\pi$, and if the equation has regions that are "UNDEFINED", try to ignore it.

### Maximize the integral of an implicit function with two parameters

Suppose we have an equation: $${(r\cos{(t)}+u})^{2}+{(-(r\sin{(t)}+v)^3+1)}^{\frac{2}{3}}=1$$

Where $x=r\cos{t}$ and $y=r\sin{t}$

The graph of this implicit relation has two regions, where one part is above the x-axis, and the other part is below the x-axis.

Suppose we're are trying to find the area of the relation above the x-axis, between the x-values ${0}\le{x}\le{2\pi}$. How can one solve for the values of u and v that can give the highest area for this equation that's is above the x-axis, so that u and v as point (u,v) is still inside the implicit relation $$(1-x^2)^3=(1-y^3)$$.

Note: The blue stripes should stop at $2\pi$, and if the equation has regions that are "UNDEFINED", try to ignore it.

### Maximize the integral of an implicit function relation with two parameters

Suppose we have an equation: $${(r\cos{(t)}+u})^{2}+{(-(r\sin{(t)}+v)^3+1)}^{\frac{2}{3}}=1$$

Where $x=r\cos{t}$ and $y=r\sin{t}$

The graph of this implicit relation has two regions, where one part is above the x-axis, and the other part is below the x-axis.

Suppose we're are trying to find the area of the relation above the x-axis, between the x-values ${0}\le{x}\le{2\pi}$. How can one solve for the values of u and v that can give the highest area for this equation that's is above the x-axis, so that u and v as point (u,v) is still inside the implicit relation $$(1-x^2)^3=(1-y^3)$$.

Note: The blue stripes should stop at $2\pi$, and if the equation has regions that are "UNDEFINED", try to ignore it.