Advertisements
Advertisements
प्रश्न
Find the area of the region bounded by the curve xy − 3x − 2y − 10 = 0, x-axis and the lines x = 3, x = 4.
उत्तर
We have,
\[xy - 3x - 2y - 10 = 0\]
\[ \Rightarrow xy - 2y = 3x + 10\]
\[ \Rightarrow y\left( x - 2 \right) = 3x + 10\]
\[ \Rightarrow y = \frac{3x + 10}{x - 2}\]
Let A represent the required area:
\[\Rightarrow A = \int_3^4 \left| y \right| d x\]
\[ = \int_3^4 \frac{3x + 10}{x - 2} d x\]
\[ = \int_3^4 \frac{3x - 6 + 16}{x - 2} d x\]
\[ = \int_3^4 \left( 3 + \frac{16}{x - 2} \right) d x\]
\[ = \left[ 3x + 16 \log \left| x - 2 \right| \right]_3^4 \]
\[ = \left[ 12 + 16 \log \left| 2 \right| - 9 - 16 \log \left| 1 \right| \right]\]
\[ = 3 + 16 \log 2\text{ sq . units }\]
APPEARS IN
संबंधित प्रश्न
Find the area of the region bounded by the curve y = sinx, the lines x=-π/2 , x=π/2 and X-axis
Using integration, find the area bounded by the curve x2 = 4y and the line x = 4y − 2.
Using integration, find the area of the region bounded by the lines y = 2 + x, y = 2 – x and x = 2.
Using the method of integration find the area of the region bounded by lines: 2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0
Area bounded by the curve y = x3, the x-axis and the ordinates x = –2 and x = 1 is ______.
Draw a rough sketch of the curve and find the area of the region bounded by curve y2 = 8x and the line x =2.
Sketch the graph of y = |x + 4|. Using integration, find the area of the region bounded by the curve y = |x + 4| and x = –6 and x = 0.
Using integration, find the area of the region bounded by the line y − 1 = x, the x − axis and the ordinates x= −2 and x = 3.
Draw a rough sketch to indicate the region bounded between the curve y2 = 4x and the line x = 3. Also, find the area of this region.
Draw a rough sketch of the graph of the function y = 2 \[\sqrt{1 - x^2}\] , x ∈ [0, 1] and evaluate the area enclosed between the curve and the x-axis.
Using integration, find the area of the region bounded by the line 2y = 5x + 7, x-axis and the lines x = 2 and x = 8.
Draw a rough sketch of the curve \[y = \frac{x}{\pi} + 2 \sin^2 x\] and find the area between the x-axis, the curve and the ordinates x = 0 and x = π.
Find the area of the region bounded by y = | x − 1 | and y = 1.
Find the area enclosed by the curves 3x2 + 5y = 32 and y = | x − 2 |.
The area of the region bounded by the parabola y = x2 + 1 and the straight line x + y = 3 is given by
The area bounded by the curve y = 4x − x2 and the x-axis is __________ .
The area bounded by the curve y2 = 8x and x2 = 8y is ___________ .
Area bounded by the curve y = x3, the x-axis and the ordinates x = −2 and x = 1 is ______.
Area lying in first quadrant and bounded by the circle x2 + y2 = 4 and the lines x = 0 and x = 2, is
Using integration, find the area of the region bounded by the line x – y + 2 = 0, the curve x = \[\sqrt{y}\] and y-axis.
Find the area of the region bound by the curves y = 6x – x2 and y = x2 – 2x
The area enclosed by the ellipse `x^2/"a"^2 + y^2/"b"^2` = 1 is equal to ______.
Find the area enclosed by the curve y = –x2 and the straight lilne x + y + 2 = 0
Find the area bounded by the curve y = 2cosx and the x-axis from x = 0 to x = 2π
Area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x2 + y2 = 32 is ______.
Using integration, find the area of the region bounded between the line x = 4 and the parabola y2 = 16x.
The area of the region bounded by the line y = 4 and the curve y = x2 is ______.
If a and c are positive real numbers and the ellipse `x^2/(4c^2) + y^2/c^2` = 1 has four distinct points in common with the circle `x^2 + y^2 = 9a^2`, then
Area of the region bounded by the curve y = |x + 1| + 1, x = –3, x = 3 and y = 0 is
Area of the region bounded by the curve `y^2 = 4x`, `y`-axis and the line `y` = 3 is:
The area bounded by `y`-axis, `y = cosx` and `y = sinx, 0 ≤ x - (<pi)/2` is
The area of the region bounded by the parabola (y – 2)2 = (x – 1), the tangent to it at the point whose ordinate is 3 and the x-axis is ______.
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 y = mx (m > 0), x = 1, x = 2 and the X-axis.
