Case study Chapter 8 (Application of Integral)

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.

(i) Region bounded by the circular edge and the straight edge(line) has points of intersection as

(a) (-2, 0) and (2, 0) (b) (0, 2) and (0, -2)

(ii) Length of each vertical strip of the region bounded by the given curves is given by

(a) $x^2-4$ (b) $\sqrt{x^2-4}$

(iii) The area of vertical strip between given circular edge and stright edge, is given by

(a) $\sqrt{x^-4}$ (b) $\sqrt{4-x^2}$

(iv) The area of horizontal strip between given circular strip and the straight edge, is given by

(a) $\sqrt{4-x^2}dx$ (b) $\sqrt{4-y^2}dx$

(v) The area of the region of the floor of the lift of the building is (in square units)

(a) $4\pi$ (b) $8\pi$

Solution: (i) Answer(a)

Given curve for circle and straight line are

x² + y² = 4 ——-(i)

y = 0 ———(ii)

∴ From (i) and (ii) we have

x² = 4 ⇒ x = ±2

∴ Points of intersection are (2, 0) and (-2, 0)

(ii) Answer (c)

Given curve , is circle whose equation is

x² + y² = 4 ——-(i)

It is a circle

y² = 4 – x² ⇒ y = √(4 – x²)

and y = 0 —–(ii)

it represents x – axis

Length of vertical strip is

y = √(4 – x²)

(iii) Answer (b)

We have

Area of one vertical strip

= y.dx = √(4 – x²).dx

(iv) Answer (b)

We have

Area of hozontal strip

= x.dy

= √(4 – y²).dy

(v) Answer (c)

We have

Area of the floor $=\int_{-2}^2y.dx =2\int_0^2√(4 - x²).dx$

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

$= 4\sin^{-1}1 - 0$

$= 4\times \frac{\pi}{2}$ 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 whole question and 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)