Class 12

Math

Calculus

Area

$Two curvesC_{1}≡[f(y)]_{2/3}+[f(x)]_{1/3}=0andC_{2}≡[f(y)]_{2/3}+[f(x)]_{2/3}=12,satisfying the relation(x−y)f(x+y)−(x+y)f(x−y)=4xy(x_{2}−y_{2})$ The area bounded by the curve $C_{2}and∣x∣+∣y∣=12 $ is

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Consider the area $S_{0},S_{1},S_{2}….$ bounded by the x-axis and half-waves of the curve $y=e_{−x}sinx,wherex≥0.$ The value of $S_{0}$ is

Find the area bounded by $y_{2}≤4x,x_{2}+y_{2}≥2x,andx≤y+2$ in the first quadrant.

Consider curves $S_{1}:∣x∣ +∣y∣ =a ,S_{2}:x_{2}+y_{2}=a_{2}andS_{3}:∣x∣+∣y∣=a.Ifαis area bounded byS_{1}andS_{2},βis area bounded byS_{1}andS_{3}andγ$ is the area bounded by $S_{2}andS_{3},$ then

Let $C_{1}andC_{2}$ be the graphs of the functions $y=x_{2}andy=2x,$ respectively, where $0≤x≤1.LetC_{3}$ be the graph of a function y=f(x), where $0≤x≤1,f(0)=0.$ For a point P on $C_{1},$ let the lines through P, parallel to the axes, meet $C_{2}andC_{3}$ at Q and R, respectively (see figure). If for every position of $P(onC_{1}),$ the areas of the shaded regions OPQ and ORP are equal, determine the function f(x).

If $a(a>0)$ is the value of parameter for each of which the are of the figure bounded by the straight line $y=1+a_{4}a_{2}−ax $ and the parabola $y=1+a_{4}x_{2}+2ax+3a_{2} $ is the greatest, then the value of $a_{4}$ is ___

Let the straight line x= b divide the area enclosed by $y=(1−x)_{2},y=0,andx=0$ into two parts $R_{1}(0≤x≤b)andR_{2}(b≤x≤1)$ such that $R_{1}−R_{2}=41 .$ Then b equals

Find the area of the region bounded by the curves $y=x+2 andy=x+11 $ between the lines x=0 and x=2.

Which of the following is the possible value/values of c for which the area of the figure bounded by the curves $y=sin2x$, the straight lines $x=π/6,x=c$ and the abscissa axis is equal to 1/2?