Question

find the volume generated by revolving the area bounded by x+y =2,x-y=4,y+3=0 about x=y

Answer 1

As the area bounded by the equations

Find-Volume generated by revolving the area bounded

Step-by-step explanation:

given equations are

x+y =2,                   equation (1)

x-y=4,                     equation (2)

y+3=0                     equation (3)

Now we find the value of x & y from the equations 1, 2,3

from equations 1 and 2 we get

$x_{1}=3$                             (A)

$y_{1}=-1$

from equations 2 and 3 we get

$x_{2} =1$                              (B)

$y_{2} =-3$

from equations 1 and 3 we get

$x_{3}=5\\y_{3}=-3$                                (C)

Now plot the graph with the use of above coordinates mentioned in the equation A,B,C.

Coordinates of the centroid (G). are

$x=\frac{x_{1}+x_{2}+x_{3} }{3} =\frac{3+1+5}{3}=3$

$y=\frac{y_{1}+y_{2}+y_{3}}{3}=\frac{-1-3-3}{3}=\frac{-7}{3}$

we know that x=y.

x-y=0            equation(4)

perpendicular distance of g from the equation 4

$d=\frac{Ax_{1}+By_{1}+C }{\sqrt{A^{2}+B^{2} } }\\d=\frac{3+\frac{7}{3} }{\sqrt{1^{2} +1^{2} } }=3.771$

Area of Δ ABC=

$\frac{1}{2} \left[\begin{array}{ccc}1&1&1\\3&1&5\\-1&-3&-3\end{array}\right] =4$

Now using Pappu's theorem

Volume generated=Area × Distance travelled by centroid

=4×2πd

$4\times2\times\frac{22}{7}\times3.771=94.813 cu.unit$

hence the value generated be 94.813 cu.unit