Case study Chapter 8 ( Application of Integral )

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)

Case study application of integral 5 — farmer has a triangular shaped field. His, son a science

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: We are given sides

y = 2x – 2 ——(i)

y = 1/2(x – 1) ——-(ii)

and y = 4 – x ——-(iii)

From (i) and (ii)

2x – 2 = 1/2(x – 1)

⇒ 4x – 4 = x – 1

⇒ 3x = 3 ⇒ x = 1

Using x = 1 in (i), we get

y = 2×1 – 2 = 2 – 2 = 0

∴ A(1 , 0)

From (i) and (iii),

2x – 2 = 4 – x

⇒ 3x = 6 ⇒ x = 2

Using x = 2 in (iii), we get

y = 4 – 2 = 2

∴ B(2, 2)

From (ii) and (iii)

4 – x = 1/2(x – 1)

⇒ 8 – 2x = x – 1

⇒ 9 = 3x

⇒ x = 3

⇒ y = 4 – x = 4 – 3 = 1

∴ C(3, 1)

(i) We have area of shaded region

$= \int_1^2\left[(2x-2)-\frac{1}{2}(x-1)\right]dx$

$= \int_1^2\left(\frac{3}{2}x-\frac{3}{2}\right)dx$

$= \frac{3}{2}\int_1^2(x-1)dx$

$= \frac{3}{2}\left[\frac{x^2}{2}-1\right]_1^2dx$

$= \frac{3}{2}[\frac{4}{2}-2-\frac{1}{2}+1]$

$= \frac{3}{2}(\frac{3}{2}-1) = \frac{3}{4}$ sq.units

(ii) We have to find area of the triangle ΔABC.

$= \int_1^2(2x-2)dx + \int_2^3(4-x)dx-\int_1^3\frac{1}{2}(x-1)$

$= 2\left[\frac{x^2}{2}-x\right]_1^2+\left[4x-\frac{x^2}{2}\right]_2^3-\frac{1}{2}\left[\frac{x^2}{2}-x\right]_1^3$

$= 2\left[2-2-\frac{1}{2}+1\right]+\left[12-\frac{9}{2}-8+2\right]-\frac{1}{2}\left[\frac{9}{2}-3-\frac{1}{2}+1\right]$

$= 1+\frc{3}{2}-1 = \frac{3}{2}$ 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 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.