Name: Area Between Two Curves - Horizontal Strip Method
Uploaded: 2019-07-19
Duration: 3 min 41 s
Description: We've learnt how to find the area bound by two curves using the vertical strip method. Can we also find the area bound by two curves using the horizontal str...

Question

Using integration, find the area of the triangle

A B C

coordinates of whose vertices are

A (4, 1), B (6, 6)

and

C (8, 4)

.

Accepted Answer

Given triangle $A B C$ is having vertices $A (4, 1), B (6, 6) and C (8, 4)$

Since we know the equation of a straight line having two points $(x_{1}, y_{1}), (x_{2}, y_{2})$ is

$\Rightarrow y - y_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} (x - x_{1})$ .

Hence, the equation of the line $A B :$ $(y - 6) = \frac{6 - 1}{6 - 4} (x - 6)$ .

$\Rightarrow A B : y = \frac{5}{2} x - 9$

The equation of $B C$ is given by $B C : (y - 4) = \frac{4 - 6}{8 - 6} (x - 8)$ .

$\Rightarrow B C : y = - x + 12$

The equation of $A C$ is given by $A C : (y - 4) = \frac{4 - 1}{8 - 4} (x - 8)$ .

$\Rightarrow A C : y = \frac{3}{4} x - 2$

$∴$ Area of $∆ A B C =$ Area of $A D B +$ Area of $B D C$ .

$= \int_{4}^{6} (y_{A B} - y_{A C}) d x + \int_{6}^{8} (y_{B C} - y_{A C}) d x$

$= \int_{4}^{6} (\frac{5}{2} x - 9 - \frac{3}{4} x + 2) d x + \int_{6}^{8} (- x + 12 - \frac{3}{4} x + 2) d x$

$= \int_{4}^{6} (\frac{7}{4} x - 7) d x + \int_{6}^{8} (- \frac{7 x}{4} + 14) d x$

$= {[\frac{7 x^{2}}{8} - 7 x]}_{4}^{6} + {[- \frac{7 x^{2}}{8} + 14 x]}_{6}^{8}$ (Apply integration)

$= [(\frac{63}{2} - 42) - (14 - 28)] + [(- 56 + 112) - (\frac{- 63}{2} + 84)]$ (Put upper and lower limits)

$= [\frac{63}{2} - 42 - 14 + 28 - 56 + 112 + \frac{63}{2} - 84]$

$= 63 - 56$

Hence, the required area is $7 sq . units$ .

Using integration, find the area of the triangle $A B C$ coordinates of whose vertices are $A (4, 1), B (6, 6)$ and $C (8, 4)$ .

Important Questions on Areas of Bounded Regions

Learn and Prepare for any exam you want !

Using integration, find the area of the triangle ABC coordinates of whose vertices are A(4, 1), B(6, 6) and C(8, 4).

Important Questions on Areas of Bounded Regions

Learn and Prepare for any exam you want !

Using integration, find the area of the triangle $A B C$ coordinates of whose vertices are $A (4, 1), B (6, 6)$ and $C (8, 4)$ .