Class 12

Math

Calculus

Area

If the area bounded by the curve $f(x)=x_{1/3}(x−1)$ and the x-axis is A, then the value of 28A is__.

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If S is the sum of cubes of possible value of c for which the area of the figure bounded by the curve $y=8x_{2}−x_{5}$, then straight lines x=1 and x=c and the abscissa axis is equal to $16/3$, then the value of [S], where [.] denotes the greatest integer function, is ___.

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}

Computing area with parametrically represented boundaries : If the boundary of a figure is represented by parametric equation, i.e., $x=x(t),y=(t),$ then the area of the figure is evaluated by one of the three formulas : $S=−α∫ β y(t)x_{′}(t)dt,$ $S=α∫ β x(t)y_{′}(t)dt,$ $S=21 α∫ β (xy_{′}−yx_{′})dt,$ Where $αandβ$ are the values of the parameter t corresponding respectively to the beginning and the end of the traversal of the curve corresponding to increasing t. If the curve given by parametric equation $x=t−t_{3},y=1−t_{4}$ forms a loop for all values of $t∈[−1,1]$ then the area of the loop is

Find the area of the region enclosed by the curve $y=∣∣ x−x1 ∣∣ (x>0)$ and the line y=2

The area enclosed by the curves $y=sinx+cosxandy=∣cosx−sinx∣$ over the interval $[0,π/2]$ is

Find the area enclosed by the figure described by the equation $x_{4}+1=2x_{2}+y_{2}.$

Find the area enclosed by $y=g(x),$ x-axis, x=1 and x=37, where g(x) is inverse of $f(x)=x_{3}+3x+1$.

Let S is the region of points which satisfies $y_{2}<16x,x<4andx_{2}−7x+12xy(x_{2}−3x+2) >0$. Its area is