Advertisements
Advertisements
प्रश्न
Using integration, find the area of the region bounded by the triangle whose vertices are (−1, 2), (1, 5) and (3, 4).
Advertisements
उत्तर
Let ABC be the required triangle with vertices A (−1, 2), B (1, 5) and C (3, 4).
Now, the equation of the side AB is
`y−2=(5−2)/(1−(−1))(x−(−1))`
`⇒ 3x − 2y + 7 = 0 ... (i)`
The equation of BC is
`y−5=(4−5)/(3−1)(x−1)`
`⇒ x + 2y − 11 = 0 ... (ii)`
The equation of AC is
`y−2=(4−2)/(3−(−1))(x−(−1))`
`⇒ x − 2y + 5 = 0 ... (iii)`
Now, area of ΔABC `= ∫_(−1)^1yABdx+∫_1^3yBCdx−∫_(−1)^3yACdx`
`=∫_(-1)^1(3x+7)/2dx + ∫_1^3(11−x)/2dx − ∫_1^3(x+5)/2dx`
`=12[((3x^2)/2+7x)_(-1)^1+(11x−x^2/2)_1^3−(x^2/2+5x)_(-1)^3]`
`=1/2[(3/2+7−3/2+7)+(33−9/2−11+12)−(9/2+15−1/2+5)]`
`= 1/2 × 8 = 4 sq. units`
APPEARS IN
संबंधित प्रश्न
Find the area of the circle 4x2 + 4y2 = 9 which is interior to the parabola x2 = 4y
Find the area bounded by curves (x – 1)2 + y2 = 1 and x2 + y 2 = 1
Using integration find the area of the triangular region whose sides have the equations y = 2x +1, y = 3x + 1 and x = 4.
Smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2 is
A. 2 (π – 2)
B. π – 2
C. 2π – 1
D. 2 (π + 2)
Area lying between the curve y2 = 4x and y = 2x is
A. 2/3
B. 1/3
C. 1/4
D. 3/4
Using the method of integration find the area bounded by the curve |x| + |y| = 1 .
[Hint: The required region is bounded by lines x + y = 1, x– y = 1, – x + y = 1 and
– x – y = 1].
Find the area bounded by curves {(x, y) : y ≥ x2 and y = |x|}.
Choose the correct answer The area of the circle x2 + y2 = 16 exterior to the parabola y2 = 6x is
A. `4/3 (4pi - sqrt3)`
B. `4/3 (4pi + sqrt3)`
C. `4/3 (8pi - sqrt3)`
D.`4/3 (4pi + sqrt3)`
Solve the following :
Find the area of the region lying between the parabolas y2 = 4x and x2 = 4y.
The area of triangle ΔABC whose vertices are A(1, 1), B(2, 1) and C(3, 3) is ______ sq.units
Find the area enclosed between the X-axis and the curve y = sin x for values of x between 0 to 2π
Find the area of the ellipse `x^2/36 + y^2/64` = 1, using integration
Find the area of the region included between y = x2 + 5 and the line y = x + 7
Find the area of the region included between the parabola y = `(3x^2)/4` and the line 3x – 2y + 12 = 0.
Calcualte the area under the curve y = `2sqrt(x)` included between the lines x = 0 and x = 1
Determine the area under the curve y = `sqrt("a"^2 - x^2)` included between the lines x = 0 and x = a.
Area lying between the curves `y^2 = 4x` and `y = 2x`
Let the area enclosed by the x-axis, and the tangent and normal drawn to the curve 4x3 – 3xy2 + 6x2 – 5xy – 8y2 + 9x + 14 = 0 at the point (–2, 3) be A. Then 8A is equal to ______.
Find the area enclosed between 3y = x2, X-axis and x = 2 to x = 3.
