Draw the rough sketch of y² = x + 1 and y² = x + 1 and determine the area enclosed by the two curves.

Using integration, find the area of the quadrant of the circle x² + y² = 4

Find the area bounded by y = x, the x - axis and the lines x = -1 and x = 2.

Calculate the area under the curve: \(y=\sqrt { 2 } x\) between the ordinates x =0 and x = 1.

Find the area lying above the x - axis and included between the circle x² + y² = 8x and the parabola y² = 4x.

Calculate the area of the region enclosed between the circles x² + y² = 1 and (x - 1)² + y² = 1.

Find the area of the region in the first quadrant enclosed by the x - axis, the line y = x and the circle x² + y² = 32.

Find the distance of the point (1, - 2, 3) from the plane x - y + z = 5, measured parallel to the line \(\frac { x-1 }{ 2 } =\frac { y-3 }{ 3 } =\frac { z+2 }{ -6 } \) 

Find the distance of the point (1, - 2, 3) from the plane x - y + z = 5 measured parallel to the line whose direction cosines are proportional to 2, 3, - 6.

Find the cartesian as well as vector equations of the planes through the intersection of planes \(\vec { r } .\left( 2\hat { i } +6\hat { j } \right) +12=0\) and   \(\vec { r } .\left( 3\hat { i } -\hat { j } +4\hat { k } \right) =0\)  which are at a unit distance from the origin.

Using integration, find the area of the triangle formed by positive x-axis and tangent and normal to the circle x + y = 4 at (1, \(\sqrt3\)).

Find the area of the region in the first quadrant enclosed by the y-axis, the line y = x and the circle x² + y² = 32, using integration.

Using integration, find the area bounded by the tangent to the curve 4y = x² at the at the point (2, 1) and the lines whose equations are x = 2y and x = 3y-3.

Using integration, find the area of the region bounded by the line 2x + y = 4, 3x - 2y = 6 and x - 3y + 5 = 0.

Using integration, find the area of the region bounded by the triangle whose vertices are : (-1, 2), (1, 5) and (3, 4).

using integration, find the area of the region bounded by the curves y = x² and y = x.

Using integration, find the area of the region enclosed between the two circles x² + y² = 9 and (x32)² + y² = 9.

Find the area enclosed by the parabola y² = 2x and the line x - y=  4.

Find the area of the region bounded by the two parabolas y² = 4ax and x² = 4ay, when a > 0.

Using integration, find the area of the region given by {(x, y) : (x² ≤ y ≤ |x| ) }