Area under the curveMedium
Question
For which of the following values of m, is the area of the region bonuded by the curve y = x - x2 and the line y = mx equals 
Options
A.-4
B.-2
C.2
D.4
Solution
Case I : When m = 0
In this case y = x - x2 .....(i)
and y = 0
are two given curves, y > 0 is total region above x-axis.
Therefore, area between y = x - x2 y = 0
is area between y = x - x2 and above the x-axis
∴
Hence, no solution exists.
Case II : When m < 0
In this case area between
y = x - x2 and y = mx is
OABCO and point of intersection are (0, 0) and
{1- m,m(1- m)}
∴ Area of curveOABCO
(given)
⇒ (1 - m)3 = 27
⇒ 1 - m = 3
⇒ m = - 2
Case III : When m > 0
In this case, y = mx and y = x - x2 intersect in (0, 0) and
{(1 - m),m(1 - m)} as shown in Fig.
Area of shaded region =
⇒
&n
In this case y = x - x2 .....(i)
and y = 0
are two given curves, y > 0 is total region above x-axis.
Therefore, area between y = x - x2 y = 0
is area between y = x - x2 and above the x-axis
∴
Hence, no solution exists.
Case II : When m < 0
In this case area between
y = x - x2 and y = mx is
OABCO and point of intersection are (0, 0) and
{1- m,m(1- m)}
∴ Area of curveOABCO
(given)⇒ (1 - m)3 = 27
⇒ 1 - m = 3
⇒ m = - 2
Case III : When m > 0
In this case, y = mx and y = x - x2 intersect in (0, 0) and
{(1 - m),m(1 - m)} as shown in Fig.
Area of shaded region =
⇒
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