calculating area using integration

calculation of area using integration (calculus method)
The area of the region bounded by y = f(x) , the x-axis , the horizontal lines x = a , x = b is given by

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The area of the region bounded by x = g(y) , the y-axis , the vertical lines y = c , y = d is given by
some problems on finding area using integration

area under one arch of the cycloid x = a(t-sint) , y =a(1-cost) ----------problem on area by integration

area between y²= x and x²=y -------------area between y²= x and x²=y

find the area of the region {(x,y) / x ² +y ² <= 1 <= x+y}
answer: (pi/4) - (1/2) more explanation on this area question

area bounded by y = x ² +2 and y = 3x ----------area between two curves
area of the cardioid r = a(1 - cosθ ) in polar form ----------- area in polar form

revision of some math topics for class XII (12th) ISC , CBSE

find the area of the region {(x,y) / x ² +y ² <= 1 <= x+y}
answer: (pi/4) - (1/2) 
evaluate integral of (x ² +3x) with limits 1 to 3 using summation (limit of a sum)
answer: (62/3)
evaluate integral of [ x*exp(x)] / (x+1) ² dx
answer: exp(x) / (x+1) + C 

∫ { [log x] / x² } dx log x refers to lnx
answer: -(1+ln x ) /x +C 
∫ dx / [ (x-2) (1 + x²)]
answer:(1/5)ln(x-2) -(1/10)ln(1 + x²) -(2/5)arctan(x)+C

problem on matrices

answer: (x= -1)


problem on determinant



solve x+y = 2 ; 2y-z = 0 ; -x-y+z = -1 using cramer's rule (using determinants)
answer : x = 3/2 , y =1/2 , z=1 
solve 3x+4y = 17 , -4x+3y =44 using matrix inversion
answer: x = -5; y =8 

solve cos ² x (dy/dx) + y = tanx
answer : y = tanx - 1 + C exp(-tanx) 
ntegral of ln(sin x) from 0 to pi/2
answer: (pi/2)ln(1/2) for explanation: 

.∫ (x²) sin x dx
answer: -x² cosx +2xsinx+2cosx+C ;explanation:

if y= tanֿ¹ { √ [(1-x)/(1+x)] } find dy/dx
answer: -1 / sqrt (1-x²) ;explanation

show that the semivertical angle of a right circular cone
of maximum volume and given slant height is tan ֿ¹(√2 )


show that the height of a closed cylinder of given volume and minimum surface area
is equal to its diameter

There is a figure (norman window) in which a rectangle is surmounted by a semicircle with diameter along one side of the rectangle .If the perimeter is given find the radius of the semicircle if the area is to be maximum (maximum amount of light is to be admitted into the room)


find the equation of the tangent at t = pi/3 on the curve x=2cost , y = 3sint
answer: 3x + 2ysqrt(3) =






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