Class 12

Math

Calculus

Area

The area enclosed by the curve $y=4−x_{2} ,y≥2 sin(22 xπ $, and the x-axis is divided by the y-axis in the ratio

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If $a(a>0)$ is the value of parameter for each of which the are of the figure bounded by the straight line $y=1+a_{4}a_{2}−ax $ and the parabola $y=1+a_{4}x_{2}+2ax+3a_{2} $ is the greatest, then the value of $a_{4}$ is ___

Let S be the area bounded by the curve $y=sinx(0≤x≤π)$ and the x-axis and T be the area bounded by the curves $y=sinx(0≤x≤2π ),y=acosx(0≤x≤2π ),$ and the x-axis $(wherea∈R_{+})$. The value of (3a) such that $S:T=1:31 $ is___.

The value of $a(a>0)$ for which the area bounded by the curves $y=6x +x_{2}1 ,y=0,x=a,andx=2a$ has the least value is ___.

If the area of the region ${(x,y):0≤y≤x_{2}+1,0≤y≤x+1,0≤x≤2}$ is A then the value of 3A-17 is___.

If the area enclosed by curve $y=f(x)andy=x_{2}+2$ between the abscissa $x=2andx=α,α>2,is(α_{3}−4α_{2}+8)$ sq. units then find function f(x). It is known that curve y=f(x) lies below the parabola $y=x_{2}+2.$

The area enclosed between the curve $y=sin_{2}xandy=cos_{2}x$ in the interval $0≤x≤π$ is

Consider curves $y=x_{2}1 ,y=4(x−1)1 .Letα$ be the value of $a(a>2)$ for which area bounded by curves between $x=2andx=ais1/aise_{2}+1andβbe the ofb∈(1,2),$ for which the area bounded by curves between x=b and $x=2is1−b1 ,$ then

Suppose $y=f(x)andy=g(x)$ are two functions whose graphs intersect at the three point (0, 4), (2,2) and (4, 0) with f(x) gt g(x) for 0 lt x lt 2 and f(x) lt g(x) for 2 lt x lt 4. If $∫_{0}[f(x)−g(x)]dx=10and∫_{2}[g(x)−f(x)]dx=5$, the area between two curves for 0 lt x lt 2, is