Class 12

Math

Calculus

Area

The area (in sq. units) of the region ${(x,y):y_{2}≥2xandx_{2}+y_{2}≤4x,x≤0,y≥0}$ is

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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

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

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

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 region bounded by an are of the cycloid $x=a(t−sint),y=a(1−cost)$ and the x-axis is

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?

If $(a,0)$, agt 0, is the point where the curve $y=sin2x−3 sinx$ cuts the x-axis first, A is the area bounded by this part of the curve, the origin and the positive x-axis. Then

Find the area bounded by $y=−x_{3}+x_{2}+16xandy=4x$

If the area bounded by $f(x)=tanx ,y=f(c),x=0andx=a,0<c<a<2π $ is minimum then find the value of c.