Advertisements
Advertisements
प्रश्न
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.
उत्तर
We have,
y = | x − 5 | intersect x = 0 and x = 1 at (0, 5) and (1, 4)
Now,
\[y = \left| x - 5 \right|\]
\[ = - \left( x - 5 \right)\text{ For all }x \in \left( 0, 1 \right)\]
Integration represents the area enclosed by the graph from x = 0 to x = 1
\[A = \int_0^1 \left| y \right| d x\]
\[ = \int_0^1 \left| x - 5 \right| d x\]
\[ = \int_0^1 - \left( x - 5 \right) d x\]
\[ = - \int_0^1 \left( x - 5 \right) d x\]
\[ = - \left[ \frac{x^2}{2} - 5x \right]_0^1 \]
\[ = - \left[ \left( \frac{1}{2} - 5 \right) - \left( 0 - 0 \right) \right]\]
\[ = - \left( - \frac{9}{2} \right)\]
\[ = \frac{9}{2}\text{ sq . units }\]
APPEARS IN
संबंधित प्रश्न
Find the area of the region bounded by the curve x2 = 16y, lines y = 2, y = 6 and Y-axis lying in the first quadrant.
Find the area lying above the x-axis and under the parabola y = 4x − x2.
Sketch the graph of y = \[\sqrt{x + 1}\] in [0, 4] and determine the area of the region enclosed by the curve, the x-axis and the lines x = 0, x = 4.
Find the area of the region bounded by the curve xy − 3x − 2y − 10 = 0, x-axis and the lines x = 3, x = 4.
Show that the areas under the curves y = sin x and y = sin 2x between x = 0 and x =\[\frac{\pi}{3}\] are in the ratio 2 : 3.
Find the area of the region common to the parabolas 4y2 = 9x and 3x2 = 16y.
Find the area of the region \[\left\{ \left( x, y \right): \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \leq \frac{x}{a} + \frac{y}{b} \right\}\]
Find the area, lying above x-axis and included between the circle x2 + y2 = 8x and the parabola y2 = 4x.
Find the area enclosed by the curve \[y = - x^2\] and the straight line x + y + 2 = 0.
Using the method of integration, find the area of the region bounded by the following lines:
3x − y − 3 = 0, 2x + y − 12 = 0, x − 2y − 1 = 0.
Find the area of the circle x2 + y2 = 16 which is exterior to the parabola y2 = 6x.
Using integration find the area of the region bounded by the curves \[y = \sqrt{4 - x^2}, x^2 + y^2 - 4x = 0\] and the x-axis.
Find the area of the region between the parabola x = 4y − y2 and the line x = 2y − 3.
Find the area of the region bounded by the parabola y2 = 2x and the straight line x − y = 4.
The area bounded by y = 2 − x2 and x + y = 0 is _________ .
The area bounded by the curve y = x4 − 2x3 + x2 + 3 with x-axis and ordinates corresponding to the minima of y is _________ .
The area bounded by the parabola y2 = 4ax, latusrectum and x-axis is ___________ .
The area bounded by the y-axis, y = cos x and y = sin x when 0 ≤ x ≤ \[\frac{\pi}{2}\] is _________ .
Area lying in first quadrant and bounded by the circle x2 + y2 = 4 and the lines x = 0 and x = 2, is
Find the equation of the parabola with latus-rectum joining points (4, 6) and (4, -2).
Find the area of the region bounded by the parabolas y2 = 6x and x2 = 6y.
The area enclosed by the circle x2 + y2 = 2 is equal to ______.
Find the area of the region included between y2 = 9x and y = x
Find the area of region bounded by the line x = 2 and the parabola y2 = 8x
Draw a rough sketch of the given curve y = 1 + |x +1|, x = –3, x = 3, y = 0 and find the area of the region bounded by them, using integration.
The area of the region bounded by the curve y = `sqrt(16 - x^2)` and x-axis is ______.
The area of the region bounded by parabola y2 = x and the straight line 2y = x is ______.
Smaller area bounded by the circle `x^2 + y^2 = 4` and the line `x + y = 2` 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 the curve y = y(x) be the solution of the differential equation, `("dy")/("d"x) = 2(x + 1)`. If the numerical value of area bounded by the curve y = y(x) and x-axis is `(4sqrt(8))/3`, then the value of y(1) is equal to ______.
The area of the region S = {(x, y): 3x2 ≤ 4y ≤ 6x + 24} is ______.
Let g(x) = cosx2, f(x) = `sqrt(x)`, and α, β (α < β) be the roots of the quadratic equation 18x2 – 9πx + π2 = 0. Then the area (in sq. units) bounded by the curve y = (gof)(x) and the lines x = α, x = β and y = 0, is ______.
Let P(x) be a real polynomial of degree 3 which vanishes at x = –3. Let P(x) have local minima at x = 1, local maxima at x = –1 and `int_-1^1 P(x)dx` = 18, then the sum of all the coefficients of the polynomial P(x) is equal to ______.
The area (in square units) of the region bounded by the curves y + 2x2 = 0 and y + 3x2 = 1, is equal to ______.
Using integration, find the area of the region bounded by line y = `sqrt(3)x`, the curve y = `sqrt(4 - x^2)` and Y-axis in first quadrant.
Sketch the region bounded by the lines 2x + y = 8, y = 2, y = 4 and the Y-axis. Hence, obtain its area using integration.
Hence find the area bounded by the curve, y = x |x| and the coordinates x = −1 and x = 1.
