Class 12

Math

Calculus

Area

The area bounded by the curves $y=cos_{−1}x,y=sin_{−1}xandy=−πx_{3},$ where $−1≤x≤1$,is

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Consider $f(x)={cosx(2π −x)_{2} 0≤x<2π 2π ≤x<π $ such that f is periodic with period $π$. Then which of the following is not true?

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. The area of the loop described as $x=3t (6−t),y=8t_{2} (6−t)$ is

If S is the sum of possible 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 $1/2,$ then the value of $π/S$ is__.

If the area enclosed by the curve $y=x andx=−y ,$ the circle $x_{2}+y_{2}=2$ above the x-axis is A, then the value of $π16 $A is__.

Find the area of the region bounded by the x-axis and the curves defined by $y=tanx(where−3π ≤x≤3π )andy=cotx(where6π ≤x≤23π ).$

Match the following lists :

If y=f(x) is a monotonic function in (a,b), then the area bounded by the ordinates at $x=a,x=b,y=f(x)andy=f(c)(wherec∈(a,b))is minimum whenc=2a+b $. $Proof :A=a∫ c (f(c)−f(x))dx+c∫ b (f(c))dx$ $=f(c)(c−a)−a∫ c (f(x))dx+a∫ b (f(x))dx−f(c)(b−c)$ $⇒A=[2c−(a+b)]f(c)+c∫ b (f(x))dx−a∫ c (f(x))dx$ Differentiating w.r.t. c, we get $dcdA =[2c−(a+b)]f_{′}(c)+2f(c)+0−f(c)−(f(c)−0)$ For maxima and minima , $dcdA =0$ $⇒f_{′}(c)[2c−(a+b)]=0(asf_{′}(c)=0)$ Hence, $c=2a+b $ $Also forc<2a+b ,dcdA <0and forc>2a+b ,dcdA >0$ Hence, A is minimum when $c=2a+b $. If the area enclosed by $f(x)=sinx+cosx,y=a$ between two consecutive points of extremum is minimum, then the value of a is

Area enclosed by the curve $y=f(x)$ defined parametrically as $x=1+t_{2}1−t_{2} ,y=1+t_{2}2t $ is equal to