#### To determine

**To find:**

The value of *c* such that the area of the region bounded by the parabolas *y=x2-c2 and y=c2-x2* is *576*

#### Answer

*6*

#### Explanation

**1) Concept:**

Put the two functions equal to each other, and find the point of intersection. These points are the integral limits.

**2) Given:**

The area of the region bounded by the parabolas *y=x2-c2 and y=c2-x2* is *576*.

**3) Calculation:**

To find the values of *c* such that the area of the region bounded by the parabolas *y=x2-c2 and y=c2-x2* is *576:*

Put the two functions equal to each other, and find the point of intersection.

Given that

*y=x2-c2, y=c2-x2*

Equate the two functions.

*x2-c2=c2-x2*

By simplification,

*2x2=2c2*

By square root,

*x=±c*

Area is the integral of the difference of two functions.

Area *=∫-cc(c2-x2-x2+c2)dx*

Since the functions are even,

*∫-ccf(x)dx=2∫0cf(x)dx*

Thus,

Area *=2∫0c(2c2-2x2)dx*

Solve integral.

*=2-23x3+2c2x0c*

Substitute limits.

*=2-23c3+2c2c-0*

By simplification,

Area *=83c3*

Given that the area of the region bounded by the parabolas *y=x2-c2 and y=c2-x2* is *576,*

By substituting *576* in the expression for area,

*83c3=576*

By simplification,

*c=6*

Therefore, the value of *c* such that the area of the region bounded by the parabolas *y=x2-c2 and y=c2-x2* is *576* is 6.

**Conclusion:**

If value of *c=6* then the area of the region bounded by the parabolas *y=x2-c2 and y=c2-x2* is *576*.