Case study Chapter 8 (Application of integral )

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.

(i) Point of intersection of the parabola and straight line are

(a) (4, 0) and (-4, 0) (b) (4, -4) and (4, 0)

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

(a) $2\sqrt{x}$ (b) $2\sqrt{y}$

(iii) Length of each vertical strip is given by

(a) $\frac{1}{4}(16-x^2)$ (b) $\frac{x^4}{4}$

(iv) Area of region bounded by parabola $x^2=4y$ and line y = 4 is (in square units)

(a) $\frac{16}{3}$ (b) 32/3

(v) Area of each vertical strip is given by

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

Solution: (i) Answer (c)

Given equation of parabola is

x² = 4 y —–(i)

And equation of straight line y = 4

∴ From (i), we get

x² = 4×4 = 16

⇒ x = ± 4

∴ Point of intersection are (4, 4) and (-4, 4)

(ii) Answer (c)

We have,

x² = 4 y —(i)

⇒ x = 2√y

Length of horizontal strip be = 2×2√y = 4√y

(iii) Answer (a)

We have,

Length of vertical strip $= 4 - \frac{x^2}{2}$

$= \frac{1}{4}(16-x^2)$

(iv) Answer (c)

We have,

Area of required bounded region

$= 2\int_0^4 x dy$

$= 2\int_0^4 2√y dy$

$= 4\left[\frac{2}{3}y^{3/2}\right]_0^4$

$= \frac{8}{3}[(4)^{3/2}-0]$ .

$= \frac{64}{3}$ sq.units

(v) Answer (b)

Area of each(one) vertical strip

= y dx

$= 4 dx -\frac{x^2}{4} dx$

$= (4 -\frac{x^2}{4}) dx$

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 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)