Advertisements
Advertisements
प्रश्न
Using the method of integration, find the area of the region bounded by the lines 3x − 2y + 1 = 0, 2x + 3y − 21 = 0 and x − 5y + 9 = 0
उत्तर
`3"x" - 2"y" + 1 = 0 ⇒ "y"_1 =((3"x"+1))/2` ...........(i)
`2"x" - 3"y" - 21 = 0 ⇒ "y"_2 =((21-2"x"))/3` .....(ii)
`"x" - 5"y" + 9 = 0 ⇒ "y"_3 = (("x"+9))/5` ......(iii)
Point of intersection of (i) and (ii) is A(3, 5)
Point of intersection of (ii) and (iii) is B(6, 3) and
Point of intersection of (iii) and (i) is C(1, 2).
Therefore, the area of the region bounded=`int_1^3"y"_1."dx"+int_3^6"y"_2."dx"-int_1^6"y"_3". dx"`
`=int_3^1((3"x"+1))/2."dx"+int_3^6((21-2"x"))/3."dx"-int_1^6(("x"+9))/5."dx"`
`=1/2((3"x"^2)/2+"x")_1^3+1/3(21"x"-"x"^2)_3^6-1/5("x"^2/2+9"x")_1^6`
`1/2[14]+1/3[36]-1/5[125/2]`
`= 7+12-12.5`
`= 6.5 "sq.units"`
APPEARS IN
संबंधित प्रश्न
Find the area of the sector of a circle bounded by the circle x2 + y2 = 16 and the line y = x in the ftrst quadrant.
Area bounded by the curve y = x3, the x-axis and the ordinates x = –2 and x = 1 is ______.
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.
Draw a rough sketch to indicate the region bounded between the curve y2 = 4x and the line x = 3. Also, find the area of this region.
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.
Calculate the area of the region bounded by the parabolas y2 = x and x2 = y.
Find the area of the region common to the parabolas 4y2 = 9x and 3x2 = 16y.
Prove that the area in the first quadrant enclosed by the x-axis, the line x = \[\sqrt{3}y\] and the circle x2 + y2 = 4 is π/3.
Find the area enclosed by the parabolas y = 5x2 and y = 2x2 + 9.
Find the area of the region bounded by the parabola y2 = 2x + 1 and the line x − y − 1 = 0.
Sketch the region bounded by the curves y = x2 + 2, y = x, x = 0 and x = 1. Also, find the area of this region.
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 enclosed between the two curves x2 + y2 = 9 and (x − 3)2 + y2 = 9.
The area bounded by the curve y = loge x and x-axis and the straight line x = e is ___________ .
The area bounded by the curves y = sin x between the ordinates x = 0, x = π and the x-axis is _____________ .
The area bounded by the parabola y2 = 4ax, latusrectum and x-axis is ___________ .
The closed area made by the parabola y = 2x2 and y = x2 + 4 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 ________ .
Area bounded by parabola y2 = x and straight line 2y = x is _________ .
The area bounded by the curve y = 4x − x2 and the x-axis is __________ .
The area of the region (in square units) bounded by the curve x2 = 4y, line x = 2 and x-axis is
Find the equation of the standard ellipse, taking its axes as the coordinate axes, whose minor axis is equal to the distance between the foci and whose length of the latus rectum is 10. Also, find its eccentricity.
Find the area of the region bounded by the parabola y2 = 2x and the straight line x – y = 4.
Find the area of the region included between y2 = 9x and y = x
The area of the region bounded by the curve x = 2y + 3 and the y lines. y = 1 and y = –1 is ______.
Area of the region bounded by the curve y = |x + 1| + 1, x = –3, x = 3 and y = 0 is
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 : [–2, 3] `rightarrow` [0, ∞) be a continuous function such that f(1 – x) = f(x) for all x ∈ [–2, 3]. If R1 is the numerical value of the area of the region bounded by y = f(x), x = –2, x = 3 and the axis of x and R2 = `int_-2^3 xf(x)dx`, then ______.
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 ______.
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.
