Advertisements
Advertisements
Question
Find the area of the region enclosed by the parabola x2 = y and the line y = x + 2
Solution
Here, x2 = y and y = x + 2
∴ x2 = x + 2
⇒ x2 – x – 2 = 0
⇒ x2 – 2x + x – 2 = 0
⇒ x(x –2) + 1(x – 2) = 0
⇒ (x – 2)(x + 1) = 0
∴ x = –1, 2
Graph of y = x + 2
| x | 0 | –2 |
| y | 2 | 0 |
Area of the required region
= `int_(-1)^2 (x + 2)"d"x - int_(-1)^2 x^2 "d"x`
= `[x^2/2 + 2x]_(-1)^2 - 1/3[x^3]_-1^2`
= `[(4/2 + 4) - (1/2 - 2)] - 1/3 [8 - (-1)]`
= `(6 + 3/2) - 1/3(9)`
= `15/2 - 3`
= `9/2` sq.units
Hence, the required area = `9/2` sq.units
APPEARS IN
RELATED QUESTIONS
triangle bounded by the lines y = 0, y = x and x = 4 is revolved about the X-axis. Find the volume of the solid of revolution.
Prove that the curves y2 = 4x and x2 = 4y divide the area of square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.
Using the method of integration, find the area of the triangular region whose vertices are (2, -2), (4, 3) and (1, 2).
Find the area of the region lying in the first quandrant bounded by the curve y2= 4x, X axis and the lines x = 1, x = 4
Find the equation of a curve passing through the point (0, 2), given that the sum of the coordinates of any point on the curve exceeds the slope of the tangent to the curve at that point by 5
Find the area of ellipse `x^2/1 + y^2/4 = 1`
Determine the area under the curve y = \[\sqrt{a^2 - x^2}\] included between the lines x = 0 and x = a.
Find the area bounded by the curve y = 4 − x2 and the lines y = 0, y = 3.
Find the area of the region \[\left\{ \left( x, y \right): \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \leq \frac{x}{a} + \frac{y}{b} \right\}\]
Find the area of the region bounded by \[y = \sqrt{x}\] and y = x.
Find the area of the region in the first quadrant enclosed by x-axis, the line y = \[\sqrt{3}x\] and the circle x2 + y2 = 16.
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 area of the region bounded by the curve y = \[\sqrt{1 - x^2}\], line y = x and the positive x-axis.
In what ratio does the x-axis divide the area of the region bounded by the parabolas y = 4x − x2 and y = x2− x?
If the area bounded by the parabola \[y^2 = 4ax\] and the line y = mx is \[\frac{a^2}{12}\] sq. units, then using integration, find the value of m.
The area of the region \[\left\{ \left( x, y \right) : x^2 + y^2 \leq 1 \leq x + y \right\}\] is __________ .
The area bounded by the curve y = f (x), x-axis, and the ordinates x = 1 and x = b is (b −1) sin (3b + 4). Then, f (x) is __________ .
The area bounded by the y-axis, y = cos x and y = sin x when 0 ≤ x ≤ \[\frac{\pi}{2}\] is _________ .
Find the area of the region bounded by the curve y2 = 4x, x2 = 4y.
Find the area of the region included between y2 = 9x and y = x
Find the area bounded by the curve y = `sqrt(x)`, x = 2y + 3 in the first quadrant and x-axis.
The area of the region bounded by the curve y = x + 1 and the lines x = 2 and x = 3 is ______.
The curve x = t2 + t + 1,y = t2 – t + 1 represents
The area bounded by `y`-axis, `y = cosx` and `y = sinx, 0 ≤ x - (<pi)/2` is
The area of the region S = {(x, y): 3x2 ≤ 4y ≤ 6x + 24} is ______.
The area (in sq.units) of the region A = {(x, y) ∈ R × R/0 ≤ x ≤ 3, 0 ≤ y ≤ 4, y ≤x2 + 3x} is ______.
Using integration, find the area of the region bounded by line y = `sqrt(3)x`, the curve y = `sqrt(4 - x^2)` and Y-axis in first quadrant.
Find the area of the smaller region bounded by the curves `x^2/25 + y^2/16` = 1 and `x/5 + y/4` = 1, using integration.
Sketch the region enclosed bounded by the curve, y = x |x| and the ordinates x = −1 and x = 1.
