To determine

To find:

The volume of the solid S whose base is the region enclosed by parabola y=1-x2

And cross sections perpendicular to the x-axis are isosceles triangles with height equal to the base.

Answer

815

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) Given:

base=height

3) Calculations:

The red line segment (s) represents the base of cross section (isosceles triangle) at x.

Area of an isosceles triangle is A=12·base·height

Since, base=height, so area of cross section is

A=12base2

Now the length of the base at x denoted s(x) is the distance from x-axis to the point (x,1-x2). So s(x)=1-x2

Now area of cross section at x is.

A(x)=s(x)2 =1-x22

Therefore, to find volume, integrate A(x) from -1 to 1, so

V= ∫011-x22dx

After expanding the square term,

V=∫011-2x2+x4dx

After integrating,

V=x-2x33+x5510

V=1-23+15-0-0+0=815

Conclusion:

The volume of the solid S whose base is the region enclosed by parabola y=1-x2

and cross sections perpendicular to the x-axis are isosceles triangles with height equal to the base is

815