Advertisements
Advertisements
Question
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.
Solution
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
RELATED QUESTIONS
Find the area of the region bounded by the parabola y2 = 4ax and its latus rectum.
Using integration, find the area bounded by the curve x2 = 4y and the line x = 4y − 2.
Find the area of the region bounded by the parabola y2 = 16x and the line x = 3.
Find the area bounded by the curve y = sin x between x = 0 and x = 2π.
Find the area of ellipse `x^2/1 + y^2/4 = 1`
Find the area of the region bounded by the parabola y2 = 4ax and the line x = a.
Draw a rough sketch of the graph of the curve \[\frac{x^2}{4} + \frac{y^2}{9} = 1\] and evaluate the area of the region under the curve and above the x-axis.
Sketch the graph y = | x + 1 |. Evaluate\[\int\limits_{- 4}^2 \left| x + 1 \right| dx\]. What does the value of this integral represent on the graph?
Find the area of the region {(x, y) : y2 ≤ 8x, x2 + y2 ≤ 9}.
Find the area of the region common to the circle x2 + y2 = 16 and the parabola y2 = 6x.
Find the area enclosed by the parabolas y = 5x2 and y = 2x2 + 9.
Find the area of the region bounded by the curves y = x − 1 and (y − 1)2 = 4 (x + 1).
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 region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2= 32.
Make a sketch of the region {(x, y) : 0 ≤ y ≤ x2 + 3; 0 ≤ y ≤ 2x + 3; 0 ≤ x ≤ 3} and find its area using integration.
Find the area of the region enclosed between the two curves x2 + y2 = 9 and (x − 3)2 + y2 = 9.
If the area enclosed by the parabolas y2 = 16ax and x2 = 16ay, a > 0 is \[\frac{1024}{3}\] square units, find the value of a.
Find the area of the region bounded by the parabola y2 = 2x and the straight line x − y = 4.
If the area above the x-axis, bounded by the curves y = 2kx and x = 0, and x = 2 is \[\frac{3}{\log_e 2}\], then the value of k is __________ .
The area bounded by the parabola y2 = 4ax and x2 = 4ay is ___________ .
The closed area made by the parabola y = 2x2 and y = x2 + 4 is __________ .
The area bounded by the curve y = x |x| and the ordinates x = −1 and x = 1 is given by
Find the coordinates of a point of the parabola y = x2 + 7x + 2 which is closest to the straight line y = 3x − 3.
Find the area of the region bounded by the parabola y2 = 2px, x2 = 2py
Find the area enclosed by the curve y = –x2 and the straight lilne x + y + 2 = 0
Area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2 = 32 is ______.
Area of the region bounded by the curve y = cosx between x = 0 and x = π is ______.
The area of the region bounded by parabola y2 = x and the straight line 2y = x is ______.
The area of the region bounded by the curve y = sinx between the ordinates x = 0, x = `pi/2` and the x-axis is ______.
The area of the region bounded by the curve y = x + 1 and the lines x = 2 and x = 3 is ______.
Find the area of the region bounded by the curve `y^2 - x` and the line `x` = 1, `x` = 4 and the `x`-axis.
The area bounded by the curve `y = x|x|`, `x`-axis and the ordinate `x` = – 1 and `x` = 1 is given by
The area enclosed by y2 = 8x and y = `sqrt(2x)` that lies outside the triangle formed by y = `sqrt(2x)`, x = 1, y = `2sqrt(2)`, is equal to ______.
Area (in sq.units) of the region outside `|x|/2 + |y|/3` = 1 and inside the ellipse `x^2/4 + y^2/9` = 1 is ______.
The area (in square units) of the region bounded by the curves y + 2x2 = 0 and y + 3x2 = 1, is equal to ______.
Find the area of the following region using integration ((x, y) : y2 ≤ 2x and y ≥ x – 4).
Find the area of the minor segment of the circle x2 + y2 = 4 cut off by the line x = 1, using integration.
Using integration, find the area of the region bounded by the curve y2 = 4x and x2 = 4y.
