Find the area of the region bounded by x 2 = 4 y , y = 2, y = 4 and the y -axis in the first quadrant.
The area of the region bounded by the curve x 2 = 4 y , the lines y = 2 and y = 4 and the y-axis. Draw a graph of the curve and lines.
Figure (1)
The area bounded by the curves and lines is area ABCD.
To calculate the area, we take a horizontal strip in the region with infinitely small width, as shown in the figure above.
To find the area of the region ABCD, integrate the area of the strip.
Area of the region ABCD = ∫ 2 4 x d y (1)
The equation of the curve is x 2 = 4 y . From this equation find the value of x in terms of y and substitute in equation (1).
x 2 = 4 y x = 2 y
Substitute 2 y for x in equation (1) and integrate.
Area of the region ABCD = ∫ 2 4 2 y d y = 2 [ y 1 2 + 1 1 2 + 1 ] 2 4 = 2 [ y 3 2 3 2 ] 2 4 = 2 ( 2 3 ) [ ( 4 ) 3 2 − ( 2 ) 3 2 ]
On further simplification, we get,
Area of the region ABCD = 4 3 [ ( 8 ) − ( 2 2 ) ] = ( 32 − 8 2 3 ) sq units
Thus, the area of the region bounded by the curve x 2 = 4 y , the lines y = 2 and y = 4 and the y-axis is ( 32 − 8 2 3 ) sq units .
The area of the region bounded by the curve
Figure (1)
The area bounded by the curves and lines is area ABCD.
To calculate the area, we take a horizontal strip in the region with infinitely small width, as shown in the figure above.
To find the area of the region ABCD, integrate the area of the strip.
The equation of the curve is
Substitute
On further simplification, we get,
Thus, the area of the region bounded by the curve
