# 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 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)

(c) (0, 0) and (4, 0)               (d) (-4, 0) and (4, 0)

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

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

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

(a) (b) (c) (d) (v) The area of the region of the floor of the lift of the building is (in square units)

(a) (b) (c) (d) 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)

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)

We have Area of one vertical strip

= y.dx = √(4 – x²).dx

We have Area of hozontal strip

= x.dy

= √(4 – y²).dy

We have Area of the floor    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.

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.

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 .

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. 