Case study Chapter 8 (Application of Integral)

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.

(i) Point of intersection of the edges(lines) y + 2x – 4 = 0 and y = 0

(a) (4, 0) (b) (0, 4)

(ii) Length of each vertical strip of the sail is given by

(a) -2x + 4 (b) y + 4

(iii) The point of intersection of the edges(lines) of the sail given by the equation y + 2x – 4 = 0 and x = 0 is

(a) (2, 0) (b) (4, 0)

(iv) Length of each horizontal strip of the sail is given by

(a) (4 – y) (b) 1/2(4 – y)

(v) The area of triangular sail is ( in square units)

(a) 4 (b) 2

Solution: (i) Answer (c)

Given equation, y + 2x – 4 = 0
x = 2
Point of intersection is (2, 0)

(ii) answer (a)

We have y + 2x – 4 = 0

y = 4 – 2x

Length of vertical strip is given by

y = 4 – 2x

i.e. -2x + 4

(iii) Answer (d)

y + 2x – 4 = 0 —(i)

and x = 0 —(ii)

From equation (i) and (ii), we have

y + 0 – 4 = 0

y = 4

Point of intersection is (0, 4)

(iv) Answer (b)

We have

y + 2x – 4 = 0

2x = 4 – y

x = 1/2(4 – y)

(v) Answer (a)

we have

Area of triangular sail $= \int_0^2y dx$

$= \int_0^2(4-2x) dx$

$= \left[4x - x^2\right]_0^2$

$= 4\times 2 - (2)^2 - 0 = 8 - 4 = 4$ sq. units

Some other Case study problem

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 4: Read the following and answer the question

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.

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)