#### To determine

**To find:**

Volume of the wedge

#### Answer

V=12833

#### Explanation

**1) Concept:**

i) Let S be a solid that lies between x=a to 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=1nA(xi*)∆x=∫abA(x)dx

**2) Calculation:**

Given that the cross section is parallel to the line of intersections of two planes

So, cross section forms the rectangle

Equation of circle of radius r centered at origin is x2+y2=r2

Therefore, x=16-y2

AB is the length of the rectangle

AB=2x

2x=216-y2

To find the width, consider the triangle as shown below,

From the triangle

tan30=BCy

13=BCy

BC=y3=w

Therefore, area of rectangle is Ay=l·w

Ay=216-y2·y3

Ay=23y16-y2

The range of y is 0 to 4

V=∫abA(y)dy=∫0423y16-y2dy

V=∫04y16-y2dy

V=23·∫04y16-y2dy

Let u=16-y2

du=-2y dy

V=23·∫04u·-12 du

V=-13·∫04u du

V=-13·∫04u12 du

V=-13 23u3204

Substituting u,

V=-13 2316-y23204

V=-233 16-4232-16-0232

V=-233 1632

V=-233 -64

V=12833

So, volume of wedge V is equal to 12833

**Conclusion:**

V=12833