Advertisements
Advertisements
Question
Find the area enclosed by the curve y = –x2 and the straight lilne x + y + 2 = 0
Advertisements
Solution
We are given that y = –x2 or x2 = –y
And the line x + y + 2 = 0
Solving the two equations,
We get x – x2 + 2 = 0
⇒ x2 – x – 2 = 0
⇒ x2 – 2x + x – 2 = 0
⇒ x(x – 2) + 1(x – 2) = 0
⇒ (x – 2)(x + 1) = 0
∴ x = –1, 2
Area of the required shaded region
= `|int_-1^2 (-x - 2) "d"x - int_-1^2 - x^2 "d"x|`
⇒ `|-[x^2/2 + x]_-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|`
⇒ `|(-15 + 6)/2| = |(-9)/2|`
= `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.
Find the area bounded by the curve y = sin x between x = 0 and x = 2π.
Using integration, find the area of the region bounded between the line x = 2 and the parabola y2 = 8x.
Find the area of the region bounded by the parabola y2 = 4ax and the line x = a.
Draw a rough sketch of the graph of the curve \[\frac{x^2}{4} + \frac{y^2}{9} = 1\] and evaluate the area of the region under the curve and above the x-axis.
Determine the area under the curve y = `sqrt(a^2-x^2)` included between the lines x = 0 and x = a.
Find the area of the minor segment of the circle \[x^2 + y^2 = a^2\] cut off by the line \[x = \frac{a}{2}\]
Find the area enclosed by the curve x = 3cost, y = 2sin t.
Find the area of the region between the circles x2 + y2 = 4 and (x − 2)2 + y2 = 4.
Find the area common to the circle x2 + y2 = 16 a2 and the parabola y2 = 6 ax.
OR
Find the area of the region {(x, y) : y2 ≤ 6ax} and {(x, y) : x2 + y2 ≤ 16a2}.
Find the area of the region bounded by the curves y = x − 1 and (y − 1)2 = 4 (x + 1).
Find the area bounded by the parabola y = 2 − x2 and the straight line y + x = 0.
Using the method of integration, find the area of the region bounded by the following lines:
3x − y − 3 = 0, 2x + y − 12 = 0, x − 2y − 1 = 0.
Using integration, find the area of the triangle ABC coordinates of whose vertices are A (4, 1), B (6, 6) and C (8, 4).
Find the area of the region bounded by y = | x − 1 | and y = 1.
Make a sketch of the region {(x, y) : 0 ≤ y ≤ x2 + 3; 0 ≤ y ≤ 2x + 3; 0 ≤ x ≤ 3} and find its area using integration.
Find the area of the region bounded by the curve y = \[\sqrt{1 - x^2}\], line y = x and the positive x-axis.
Find the area of the region enclosed between the two curves x2 + y2 = 9 and (x − 3)2 + y2 = 9.
Find the area enclosed by the curves 3x2 + 5y = 32 and y = | x − 2 |.
Find the area bounded by the parabola y2 = 4x and the line y = 2x − 4 By using horizontal strips.
Find the area of the region bounded by the parabola y2 = 2x and the straight line x − y = 4.
The area bounded by y = 2 − x2 and x + y = 0 is _________ .
Find the coordinates of a point of the parabola y = x2 + 7x + 2 which is closest to the straight line y = 3x − 3.
Find the area of the region bound by the curves y = 6x – x2 and y = x2 – 2x
Using integration, find the area of the smaller region bounded by the ellipse `"x"^2/9+"y"^2/4=1`and the line `"x"/3+"y"/2=1.`
Find the area of the region bounded by the curve y = x3 and y = x + 6 and x = 0
Find the area of the region included between y2 = 9x and y = x
Find the area of region bounded by the triangle whose vertices are (–1, 1), (0, 5) and (3, 2), using integration.
The area of the region bounded by the curve x2 = 4y and the straight line x = 4y – 2 is ______.
The area of the region bounded by the curve y = `sqrt(16 - x^2)` and x-axis is ______.
The area of the region bounded by the ellipse `x^2/25 + y^2/16` = 1 is ______.
The area of the region bounded by the circle x2 + y2 = 1 is ______.
The area of the region bounded by the curve y = x + 1 and the lines x = 2 and x = 3 is ______.
The area of the region bounded by the line y = 4 and the curve y = x2 is ______.
Find the area bounded by the curve y = |x – 1| and y = 1, using integration.
Find the area of the region enclosed by the curves y2 = x, x = `1/4`, y = 0 and x = 1, using integration.
Let the curve y = y(x) be the solution of the differential equation, `("dy")/("d"x) = 2(x + 1)`. If the numerical value of area bounded by the curve y = y(x) and x-axis is `(4sqrt(8))/3`, then the value of y(1) is equal to ______.
Using integration, find the area of the region bounded by y = mx (m > 0), x = 1, x = 2 and the X-axis.
Make a rough sketch of the region {(x, y) : 0 ≤ y ≤ x2 + 1, 0 ≤ y ≤ x + 1, 0 ≤ x ≤ 2} and find the area of the region, using the method of integration.
