Advertisements
Advertisements
Question
Find the area bounded by the curve y = sinx between x = 0 and x = 2π.
Advertisements
Solution
Required area = `int_0^pi sinx "d"x + int_pi^(2pi) |sin x|"d"x`
= `-[cos x]_0^pi + |(- cos x)|_pi^(2pi)`
= `=[cos pi - cos 0] + [cos 2pi - cos pi]`
= `-[-1 - 1] + [1 + 1]`
= 2 + 2
= 4 sq.units
APPEARS IN
RELATED QUESTIONS
Find the area bounded by the curve y2 = 4ax, x-axis and the lines x = 0 and x = a.
Using the method of integration, find the area of the triangular region whose vertices are (2, -2), (4, 3) and (1, 2).
Sketch the graph of y = |x + 4|. Using integration, find the area of the region bounded by the curve y = |x + 4| and x = –6 and x = 0.
Find the area lying above the x-axis and under the parabola y = 4x − x2.
Sketch the region {(x, y) : 9x2 + 4y2 = 36} and find the area of the region enclosed by it, using integration.
Draw a rough sketch of the graph of the function y = 2 \[\sqrt{1 - x^2}\] , x ∈ [0, 1] and evaluate the area enclosed between the curve and the x-axis.
Sketch the graph y = |x + 1|. Evaluate\[\int\limits_{- 4}^2 \left| x + 1 \right| dx\]. What does the value of this 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.
Calculate the area of the region bounded by the parabolas y2 = x and x2 = y.
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.
Draw a rough sketch of the region {(x, y) : y2 ≤ 5x, 5x2 + 5y2 ≤ 36} and find the area enclosed by the region using method of integration.
Find the area of the region bounded by the curves y = x − 1 and (y − 1)2 = 4 (x + 1).
Find the area of the region bounded by y = | x − 1 | and y = 1.
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 {(x, y): x2 + y2 ≤ 4, x + y ≥ 2}.
Using integration, find the area of the following region: \[\left\{ \left( x, y \right) : \frac{x^2}{9} + \frac{y^2}{4} \leq 1 \leq \frac{x}{3} + \frac{y}{2} \right\}\]
Find the area enclosed by the curves y = | x − 1 | and y = −| x − 1 | + 1.
Find the area bounded by the parabola y2 = 4x and the line y = 2x − 4 By using horizontal strips.
The area bounded by the curve y = loge x and x-axis and the straight line x = e is ___________ .
Area bounded by parabola y2 = x and straight line 2y = x is _________ .
The area bounded by the curve y2 = 8x and x2 = 8y is ___________ .
The area bounded by the y-axis, y = cos x and y = sin x when 0 ≤ x ≤ \[\frac{\pi}{2}\] is _________ .
Find the coordinates of a point of the parabola y = x2 + 7x + 2 which is closest to the straight line y = 3x − 3.
Using integration, find the area of the region bounded by the line x – y + 2 = 0, the curve x = \[\sqrt{y}\] and y-axis.
Find the area of the region bounded by the parabola y2 = 2x and the straight line x – y = 4.
The area of the region bounded by the curve x = y2, y-axis and the line y = 3 and y = 4 is ______.
The area of the region bounded by the curve y = x2 + x, x-axis and the line x = 2 and x = 5 is equal to ______.
Find the area of the region bounded by y = `sqrt(x)` and y = x.
Compute the area bounded by the lines x + 2y = 2, y – x = 1 and 2x + y = 7.
The area of the region bounded by the curve y = `sqrt(16 - x^2)` and x-axis is ______.
Area of the region bounded by the curve y = cosx between x = 0 and x = π is ______.
Using integration, find the area of the region `{(x, y): 0 ≤ y ≤ sqrt(3)x, x^2 + y^2 ≤ 4}`
Area of the region bounded by the curve `y^2 = 4x`, `y`-axis and the line `y` = 3 is:
Find the area of the region bounded by `x^2 = 4y, y = 2, y = 4`, and the `y`-axis in the first quadrant.
Using integration, find the area of the region bounded by the curves x2 + y2 = 4, x = `sqrt(3)`y and x-axis lying in the first quadrant.
Let f(x) be a non-negative continuous function such that the area bounded by the curve y = f(x), x-axis and the ordinates x = `π/4` and x = `β > π/4` is `(βsinβ + π/4 cos β + sqrt(2)β)`. Then `f(π/2)` is ______.
Let a and b respectively be the points of local maximum and local minimum of the function f(x) = 2x3 – 3x2 – 12x. If A is the total area of the region bounded by y = f(x), the x-axis and the lines x = a and x = b, then 4A is equal to ______.
Using integration, find the area bounded by the curve y2 = 4ax and the line x = a.
Hence find the area bounded by the curve, y = x |x| and the coordinates x = −1 and x = 1.
