Class 12

Math

Calculus

Area

The area bounded by the curves y=cos x and y= sin x between the ordinates x=0 and $x=3π/2$ is

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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 bounded by $f(x)=3x_{3} −x_{2}+a$ and the straight lines x=0, x=2, and the x-axis is minimum, then the value of a is

A farmer $F_{1}$ has a land in the shape of a triangle with vertices at $P(0,0),Q(1,1)andR(2,0).$ From this land, a neighboring farmer $F_{2}$ takes away the region which lies between the side PQ and curve of the from $y=x_{n}(n>1).$ If the area of the region taken away by the farmer $F_{2}$ is exactly $30%$ of the area of $ΔPQR$, then the value of n is ___.

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

If $S_{0},S_{1},S_{2},…$ are areas bounded by the x-axis and half-wave of the curve $y=sinπx ,then prove thatS_{0},S_{1},S_{2},…$ are in A.P…

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

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__.

The ratio of the areas of two regions of the curve $C_{1}≡4x_{2}+π_{2}y_{2}=4π_{2}$ divided by the curve $C_{2}≡y=−(sgn(x−2π ))cosx$ (where sgn (x) = signum (x)) is

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