# Case study Chapter 8 ( Application of integral )

**Case study 4: Read the following and answer the question(Case study application of integral 4)**

A boy design a pizza by cutting it with a knife on a card board. If pizza is circular in shape which is represented by the

equation and edge of knife represents a straight line given by .

**(i) Point of intersection of the line and pizza shown in figure is(are)**

(a) (1, √3) (b) (√3, 1) and (-√3, -1)

(c) (2, 0) (d) (2, 0) and (0, 0)

**(ii) The area of piece OABO of a cicular design pizza is**

(a) π sq.units (b) 2π sq.units

(c) π/3 sq.units (d) 2π/3 sq.units

**(iii) The area of the piece OABCO of the circular designed pizza is**

(a) π sq.units (b) 2π sq.units

(c) 3π sq.units (d) 4π sq.units

**(iv) The area of semicircular shape of the pizza is**

(a) π sq.units (b) 2π sq.units

(c) 4π sq.units (d) 8π sq.units

**(v) Area of the whole pizza is (in sq. units)**

(a) π (b) 2π

(c) 4π (d) 8π

**Solution:(i) Answer(b)**

We have ,

x² + y² = 4 —-(i)

and x = √3y —-(ii)

Solving (i) and (ii) we get,

3y² + y² = 4

⇒ 4y² = 4

⇒ y² = 1 ⇒ y = ±1

and x = ±√3

**Point of intersection are (√3, 1) and (-√3, -1)**

**(ii) Answer (c)**

Required area given by,

sq.units

**(iii) Answer (a)**

We have area of OABCO

sq.units

**(iv) Answer (b)**

Required area = 2×Area of OABCO

square units

**(v) Answer (c)**

Area of whole pizza = 2× Area of semicicular shape

sq.units

## Some other Case study problem

**Case study 1: Read the following and answer the question.(Case study application of integral 1)**

Nowadays, almost every boat has a triangular sail. By using a triangular sail design it has become possible to travel against the wind using a technique known as tacking. Tacking allows the boat to travel forward with r

triangular sail on the walls and three edges(lines) at the triangular sail are given by the equation x = 0, y = 0 and y + 2x – 4 = 0 respectively.

**Solution:** For whole question and solution click here

**Case study 2: Read the following and answer the question(Case study application of integral 2)**

An architect designs a building whose lift (elevator ) is from outside of the building attached to the walls. The floor (base ) of the lift (elevator) is in semicircular shape. The floor of the elevator (lift) whose circular edge is given by the equation and the straight edge(line) is given by the equation y = 0.

**Solution:** For solution click here

**Case study 3: Read the following and answer the question(Case study application of integral 3)**

A student designs an open air Honeybee nest on the branch of a tree, whose plane figure is parabolic and the branch of tree is given by a straight line.

**Solution:** For solution click here

**Case study 5:-A farmer has a triangular shaped field. His, son a science student observes the triangular field has three edges and can be drawn on a plain paper with three lines given by its equations.(Case study application of integral 5)**

**Based on the above information answer the following question:**

**(i) Find the area of the shaped region in the figure shown below.**

**(ii) Find the area of the triangle ΔABC.**

**Solution:** For solution click here