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 $x^2+y^2=4$ and edge of knife represents a straight line given by $x =\sqrt{3}y$ .

Case study application of integral 4 — A boy design a pizza by cutting it with a knife on a card board.

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

(a) (1, √3) (b) (√3, 1) and (-√3, -1)

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

(a) π sq.units (b) 2π sq.units

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

(a) π sq.units (b) 2π sq.units

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

(a) π sq.units (b) 2π sq.units

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

(a) π (b) 2π

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,

$= \int_0^{\sqrt{3}}\frac{x}{\sqrt{3}}dx+\int_{\sqrt{3}}^2 \sqrt{4-x^2}dx$

$=\left[\frac{x^2}{2\sqrt{3}}\right]_0^{\sqrt{3}}+\left[\frac{x}{2}\sqrt{4-x^2}+\frac{4}{2}\sin^{-1}\frac{x}{2}\right]_{\sqrt{3}}^2$

$= \frac{(\sqrt{3})^2}{2\sqrt{3}}-0+2\sin^{-1}(1)-\frac{\sqrt{3}}{2}-2\sin^{-1}\frac{\sqrt{3}}{2}$

$= \frac{\sqrt{3}}{2}+2\times \frac{\pi}{2}-\frac{\sqrt{3}}{2}-2\times \frac{\pi}{2}$

$= \pi-\frac{2\pi}{3} = \frac{\pi}{3}$ sq.units

(iii) Answer (a)

We have area of OABCO

$=\int_0^2 y.dx$

$= \int_0^2 \sqrt{4-x^2}dx$

$=\left[\frac{x}{2}\sqrt{4-x^2}+\frac{4}{2}\sin^{-1}\frac{x}{2}\right]_0^2$

$= [0+2\sin^{-1}(1)-0]$

$= 2\times \frac{\pi}{2}$

$= \pi$ sq.units

(iv) Answer (b)

Required area = 2×Area of OABCO

$= 2\pi$ square units

(v) Answer (c)

Area of whole pizza = 2× Area of semicicular shape

$= 2\times 2\pi$

$= 4\pi$ 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 $x^2+y^2 = 4$ 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)