#### To determine

**To find:**

The volume of the solid S which is bounded by circles that are perpendicular to the x-axis, intersect the x-axis and have centers on the parabola y=121-x2, -1≤x≤1.

#### Answer

415π

#### Explanation

**1) Concept:**

Definition of volume:

Let S be a solid that lies between x=a and x=b. If the cross sectional area of S in the plane Px, through x and perpendicular to the x-axis, is A(x), where A is continuous function, then the volume of S is

V=limn→∞∑i=1nAxi*∆x=∫abAxdx

**2) Calculations:**

The cross section of S at coordinate x,-1≤x≤1, is a circle centered at the

point (x,121-x2) ^ the point enterered ich is bounded by circles that are perpendicular to the , with radius 121-x2. The radius is the distance of centre from x-axis because the circle is intersecting x-axis.

The area of a circle is πr2.

Therefore, the area of cross section at x is

A(x)=π121-x22=π4(1-2x2+x4).

The volume of S is

V=∫-11π41-2x2+x4dx=2∫01π41-2x2+x4dx

After integrating

V=π2x-2x33+x5510

By using the limits

V=π21-2312+155-0=π213+15=4π15.

**Conclusion:**

The volume of the solid S which is bounded by circles that are perpendicular to the x-axis, intersect the x-axis and have centers on the parabola y=121-x2, -1≤x≤1 is 415π