Class 12

Math

Calculus

Area

If the line $x=α$ divides the area of region $R={(x,y)∈R_{2}:x_{3}≤y≤x,0≤x≤1}$ into two equal parts, then

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The area made by curve $f(x)=[x]+x−[x] $ and x-axis when $0≤x≤n(n∈N)$ is equal to { where [x] is greatest integer function}

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).

The area enclosed by the curves $xy_{2}=a_{2}(a−x)and(a−x)y_{2}=a_{2}x$ is

Area bounded by the relation $[2x]+[y]=5,x,y>0$ is__ (where [.] represents greatest integer funciton).

Consider two curves $C_{1}:y_{2}=4[y ]xandC_{2}:x_{2}=4[x ]y,where[.]$ denotes the greatest integer function. Then the area of region enclosed by these two curves within the square formed by the lines $x=1,y=1,x=4,y=4$ is

Area bounded by $f(x)=x_{2}+1x_{2}−1 $ and the line y = 1 is

The value of the parameter a such that the area bounded by $y=a_{2}x_{2}+ax+1,$ coordinate axes, and the line x=1 attains its least value is equal to

Area of region bounded by the curve $y=4+x_{2}4−x_{2} ,25y_{2}=9xandy=53 ∣x∣−56 $ which contains (1, 0) point in its interior is