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).
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 region lying between the parabolas y2 = 4ax and x2 = 4ay.
Using integration finds the area of the region bounded by the triangle whose vertices are (–1, 0), (1, 3) and (3, 2).
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)
Using integration, find the area of region bounded by the triangle whose vertices are (–2, 1), (0, 4) and (2, 3).
Show that the rectangle of the maximum perimeter which can be inscribed in the circle of radius 10 cm is a square of side `10sqrt2` cm.
Find the area included between the parabolas y2 = 4ax and x2 = 4by.
Area enclosed between the curve y2 (2a − x) = x3 and the line x = 2a above x-axis is ___________ .
Area lying between the curves y2 = 4x and y = 2x is
The area of the region included between the parabolas y2 = 16x and x2 = 16y, is given by ______ sq.units
The area enclosed between the two parabolas y2 = 20x and y = 2x is ______ sq.units
Find the area of sector bounded by the circle x2 + y2 = 25, in the first quadrant−
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 lying between the parabolas 4y2 = 9x and 3x2 = 16y
Find the area of the region included between y = x2 + 5 and the line y = x + 7
Find the area enclosed between the circle x2 + y2 = 9, along X-axis and the line x = y, lying in the first quadrant
Find the area enclosed by the curve x = 3 cost, y = 2 sint.
Find the area of the region included between the parabola y = `(3x^2)/4` and the line 3x – 2y + 12 = 0.
Find the area of a minor segment of the circle x2 + y2 = a2 cut off by the line x = `"a"/2`
Calcualte the area under the curve y = `2sqrt(x)` included between the lines x = 0 and x = 1
Draw a rough sketch of the curve y = `sqrt(x - 1)` in the interval [1, 5]. Find the area under the curve and between the lines x = 1 and x = 5.
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`
Find the area enclosed between 3y = x2, X-axis and x = 2 to x = 3.
