Advertisements
Advertisements
Question
Find the area of the region bounded by the following curves, the X-axis and the given lines: 2y + x = 8, x = 2, x = 4
Advertisements
Solution
Let A be the required area.
Consider the equation 2y + x = 8
i.e., y = `(8 - x)/(2)`
∴ A = `int_2^4 y*dx`
= `int_2^4 (8 - x)/(2)*dx`
= `(1)/(2) int_2^4 (8 - x)*dx`
= `(1)/(2)[8x - x^2/2]_2^4`
= `(1)/(2)[(8 xx 4 - 4^2/2) - (8 xx 2 - 2^2/2)]`
= `(1)/(2)(32 - 8) - (16 - 2)]`
= `(1)/(2)(24 - 14)`
= `(1)/(2) xx 10`
∴ A = 5 sq. units.
APPEARS IN
RELATED QUESTIONS
Using integration find the area of the region {(x, y) : x2+y2⩽ 2ax, y2⩾ ax, x, y ⩾ 0}.
Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle
`x^2+y^2=4 at (1, sqrt3)`
Find the area of the region bounded by the ellipse `x^2/4 + y^2/9 = 1.`
Find the area of the region in the first quadrant enclosed by x-axis, line x = `sqrt3` y and the circle x2 + y2 = 4.
Find the area of the region bounded by the parabola y = x2 and y = |x| .
Find the area bounded by the curve x2 = 4y and the line x = 4y – 2
Area of the region bounded by the curve y2 = 4x, y-axis and the line y = 3 is ______.
Find the area under the given curve and given line:
y = x2, x = 1, x = 2 and x-axis
Draw a rough sketch and find the area bounded by the curve x2 = y and x + y = 2.
Find the area of the region.
{(x,y) : 0 ≤ y ≤ x2 , 0 ≤ y ≤ x + 2 ,-1 ≤ x ≤ 3} .
Using integration find the area of the triangle formed by negative x-axis and tangent and normal to the circle `"x"^2 + "y"^2 = 9 "at" (-1,2sqrt2)`.
Find the area of the region bounded by the following curves, the X-axis and the given lines: y = `sqrt(16 - x^2)`, x = 0, x = 4
State whether the following is True or False :
The area bounded by the curve y = f(x), X-axis and lines x = a and x = b is `|int_"a"^"b" f(x)*dx|`.
Choose the correct alternative:
Area of the region bounded by y2 = 16x, x = 1 and x = 4 and the X axis, lying in the first quadrant is ______
State whether the following statement is True or False:
The area bounded by the curve y = f(x) lies on the both sides of the X-axis is `|int_"a"^"b" "f"(x) "d"x| + |int_"b"^"c" "f"(x) "d"x|`
State whether the following statement is True or False:
The equation of the area of the circle is `x^2/"a"^2 + y^2/"b"^2` = 1
The area bounded by the parabola x2 = 9y and the lines y = 4 and y = 9 in the first quadrant is ______
Find the area of the region bounded by the curve 4y = 7x + 9, the X-axis and the lines x = 2 and x = 8
Find area of the region bounded by the parabola x2 = 36y, y = 1 and y = 4, and the positive Y-axis
Find area of the region bounded by the parabola x2 = 4y, the Y-axis lying in the first quadrant and the lines y = 3
Find the area of the circle x2 + y2 = 62
If `int_0^(pi/2) log (cos x) "dx" = - pi/2 log 2,` then `int_0^(pi/2) log (cosec x)`dx = ?
Area enclosed between the curve y2(4 - x) = x3 and line x = 4 above X-axis is ______.
The area of the region bounded by the X-axis and the curves defined by y = cot x, `(pi/6 ≤ x ≤ pi/4)` is ______.
The area enclosed by the parabolas x = y2 - 1 and x = 1 - y2 is ______.
The area of the region bounded by the curve y = x2, x = 0, x = 3, and the X-axis is ______.
The area of the region bounded by the curve y = sin x and the x-axis in [–π, π] is ______.
Area bounded by y = sec2x, x = `π/6`, x = `π/3` and x-axis is ______.
The area bounded by the curve, y = –x, X-axis, x = 1 and x = 4 is ______.
For the ellipse \[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\], which expression for \[y\] is obtained before selecting the first-quadrant branch?
