Advertisements
Advertisements
Question
Find the area bounded by the curve y = |x – 1| and y = 1, using integration.
Solution
We have, y = (x – 1)
y = x – 1, if x – 1 ≥ 0
y = –x + 1, if x – 1 < 0
Required Area = Area of shaded region
A = `int_0^2 ydx`
= `int_0^1(1 - x)dx + int_1^2(x - 1)dx`
= `[x - x^2/2]_0^1 + [x^2/2 - x]_1^2`
= `(1 - 1/2) - (0 - 0/2) + (4/2 - 2) - (1/2 - 1)`
= `1/2 + 1/2`
= 1 sq.unit
APPEARS IN
RELATED QUESTIONS
Area bounded by the curve y = x3, the x-axis and the ordinates x = –2 and x = 1 is ______.
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
Sketch the graph y = | x − 5 |. Evaluate \[\int\limits_0^1 \left| x - 5 \right| dx\]. What does this value of the integral represent on the graph.
Show that the areas under the curves y = sin x and y = sin 2x between x = 0 and x =\[\frac{\pi}{3}\] are in the ratio 2 : 3.
Using integration, find the area of the region bounded by the triangle ABC whose vertices A, B, C are (−1, 1), (0, 5) and (3, 2) respectively.
Find the area of the region {(x, y) : y2 ≤ 8x, x2 + y2 ≤ 9}.
Find the area enclosed by the parabolas y = 5x2 and y = 2x2 + 9.
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 bounded by the parabola y2 = 2x + 1 and the line x − y − 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).
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 figure bounded by the curves y = | x − 1 | and y = 3 −| x |.
The area of the region formed by x2 + y2 − 6x − 4y + 12 ≤ 0, y ≤ x and x ≤ 5/2 is ______ .
The ratio of the areas between the curves y = cos x and y = cos 2x and x-axis from x = 0 to x = π/3 is ________ .
Smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2 is
Find the area of the region bounded by the curve y = x3 and y = x + 6 and x = 0
Sketch the region `{(x, 0) : y = sqrt(4 - x^2)}` and x-axis. Find the area of the region using integration.
Find the area of the region bounded by y = `sqrt(x)` and y = x.
The area of the region bounded by parabola y2 = x and the straight line 2y = x 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 ______.
Area lying in the first quadrant and bounded by the circle `x^2 + y^2 = 4` and the lines `x + 0` and `x = 2`.
Find the area of the region bounded by the ellipse `x^2/4 + y^2/9` = 1.
The area bounded by the curve `y = x|x|`, `x`-axis and the ordinate `x` = – 1 and `x` = 1 is given by
Area (in sq.units) of the region outside `|x|/2 + |y|/3` = 1 and inside the ellipse `x^2/4 + y^2/9` = 1 is ______.
The area (in sq.units) of the region A = {(x, y) ∈ R × R/0 ≤ x ≤ 3, 0 ≤ y ≤ 4, y ≤x2 + 3x} is ______.
The area (in square units) of the region bounded by the curves y + 2x2 = 0 and y + 3x2 = 1, is equal to ______.
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.
Using integration, find the area bounded by the curve y2 = 4ax and the line x = a.
