Question

Find the area bounded by curve x² = 4y & line x = 4y-2 . Explain each step & the figure ,i m nt able to understand its figure

Answer 1

x² = 4 y      and      x = 4 y - 2  ie.,,  y =  x / 4 + 1/2

area bounded by them... we find it by the integration..

the parabola x² = 4 y  has  axis as  x = 0  ie.,  y axis.  Focus is at  (0, 1) and its vertex is at origin.  It is like  a  U curve standing on the x axis at the origin.

The line intersects the x axis at (-2, 0) and  y axis at the point (0, 1/2).

these two graphs intersect at :
           x² = 4 y =  (4 y - 2)²
      =>  4 y² - 5y + 1  = 0 
      =>  y  =  1  or  1/4
     =>  x =  2  or  -1

Thus  we have to find the area bound between the two curves between the points  (2,1 ) and (-1, 1/4).

we know that the straight line is above the x axis when it cuts the curve.  y1(x) is the line. and y2 (x) is the parabola.

$area= \int\limits^2_{-1} {[y_1(x)-y_2(x)]} \, dx \\\\ =\int\limits^2_{-1} {[x/4\ +\ 1/2 -\ x^2/4]} \, dx \\\\ = [x^2/8+x/2-x^3/12 ]_{-1}^2\\\\ =[ (4/8+2/2-\ 2^3/12)\ -\ (1/8-1/2+1/12)]\\\\= 5/6+7/24\\\\=27/24=9/8$