Advertisements
Advertisements
Question
The area of the region bounded by y2 = 4x, the X-axis and the lines x = 1 and x = 4 is _______.
Options
28 sq. units
3 sq. units
`(28)/(3)` sq. units
`(3)/(28)` sq. units
Advertisements
Solution
`bb((28)/(3))` sq. units
Explanation:
The right-handed parabola in this example, y2 = 4x, has its vertex at the origin, and the lines parallel to the y-axis at x = 1 to x = 4 units distance are x = 1, x = 4.
Similarly y2 = 4x contains even power of y and is symmetrical about the x-axis.
So the required area = Area of ABCD
Area of ABCD = `int_1^4ydx=int_1^4sqrt(4x)dx`
It can be written as
= `2int_1^4sqrtxdx`
= `2[(x3/2)/(3/2)]_1^4`
= `2[(2x3/2)/3]_1^4`
Substituting the values we get
= `2((2(4)3/2)/3-(2(1)3/2)/3)`
= `4(8/3-1/3)`
= `4(7/3)`
= `28/3` sq. units
APPEARS IN
RELATED QUESTIONS
Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis.
Find the area of the region bounded by y2 = 9x, x = 2, x = 4 and the x-axis in the first quadrant.
Find the area bounded by the curve x2 = 4y and the line x = 4y – 2
Sketch the graph of y = |x + 3| and evaluate `int_(-6)^0 |x + 3|dx`
Find the area of the smaller region bounded by the ellipse `x^2/9 + y^2/4` and the line `x/3 + y/2 = 1`
Find the area of the smaller region bounded by the ellipse `x^2/a^2 + y^2/b^2 = 1` and the line `x/a + y/b = 1`
Find the area of the region {(x, y) : y2 ≤ 4x, 4x2 + 4y2 ≤ 9}
Find the equation of an ellipse whose latus rectum is 8 and eccentricity is `1/3`
Find the area of the smaller region bounded by the ellipse \[\frac{x^2}{9} + \frac{y^2}{4} = 1\] and the line \[\frac{x}{3} + \frac{y}{2} = 1 .\]
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: 2y + x = 8, x = 2, x = 4
Find the area of the region bounded by the parabola y2 = 4x and the line x = 3.
Using definite integration, area of the circle x2 + y2 = 49 is _______.
Fill in the blank :
Area of the region bounded by x2 = 16y, y = 1, y = 4 and the Y-axis, lying in the first quadrant is _______.
Solve the following :
Find the area of the region bounded by the curve xy = c2, the X-axis, and the lines x = c, x = 2c.
Solve the following :
Find the area of the region bounded by the curve y = x2 and the line y = 10.
State whether the following statement is True or False:
The area of portion lying below the X axis is negative
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
The area bounded by the X-axis, the curve y = f(x) and the lines x = 1, x = b is equal to `sqrt("b"^2 + 1) - sqrt(2)` for all b > 1, then f(x) is ______.
The area of the region bounded by the curve y = x IxI, X-axis and the ordinates x = 2, x = –2 is ______.
The area included between the parabolas y2 = 4a(x +a) and y2 = 4b(x – a), b > a > 0, is
The area (in sq. units) of the region {(x, y) : y2 ≥ 2x and x2 + y2 ≤ 4x, x ≥ 0, y ≥ 0} is ______.
Find the area of the region lying in the first quadrant and bounded by y = 4x2, x = 0,y = 2 and y = 4.
Find the area of the regions bounded by the line y = −2x, the X-axis and the lines x = −1 and x = 2.
What is the total area bounded by \[y=f(x)\], the \[x\]-axis, and the ordinates \[x=a\] and \[x=b\], when the region is above the \[x\]-axis?
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?
Which integral represents the enclosed area of the ellipse after using the positive first-quadrant value of \[y\]?
Which antiderivative is used in evaluating \[\int \sqrt{a^2-x^2}\,dx\] for the ellipse area?
