Find the greatest value of `x^2 y^3`, where x and y lie in the first quadrant on the line `3x+4y=5`.

Find the area of the region if the first quadrant bounded by the parabola y^(2)=4x , the line x+y=3 and y-axis.

Find the area of the region bounded by x^(2)= 4y, y = 2, y = 4 and the y-axis in the first quadrant.

Find the area of the region in the first quadrant enclosed by the x-aixs, the line y = x, and the circle x^(2) + y^(2) = 32 .

If sin x = 3/5, cos y =- (12)/(13), where x and y both lie in second quadrant, find the vlaue of sin (x +y).

Find the area lying in the first quadrant and bounded by the curve y=x^3 and the line y=4xdot

Find the area of the region lying in the first quadrant and bounded by y = 4x^(2) , x = 0, y = 1 and y = 4.

Find the area of the region in the first quadrant enclosed by x-axis, line x = sqrt3y and the circle x^(2) + y^(2) = 4 .

The number of integral points (x,y) (i.e, x and y both are integers) which lie in the first quadrant but not on the coordinate axes and also on the straight line 3x + 5y = 2007 is equal to

Find the point (alpha,beta) on the ellipse 4x^2+3y^2=12 , in the first quadrant, so that the area enclosed by the lines y=x ,y=beta,x=alpha , and the x-axis is maximum.