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Question
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.
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Solution
We have,
Straight line 2y = 5x + 7 intersect x-axis and y-axis at ( −1.4, 0) and (0, 3.5) respectively.
Also x = 2 and x = 8 are straight lines as shown in the figure.
The shaded region is our required region whose area has to be found.
When we slice the shaded region into vertical strips, we find that each vertical strip has its lower end on x-axis and upper end on the line
2y = 5x + 7
So, approximating rectangle shown in figure has length = y and width = dx and area = y dx.
The approximating rectangle can move from x = 2 to x = 8.
So, required is given by,
\[A = \int_2^8 y d x\]
\[ = \int_2^8 \left( \frac{5x + 7}{2} \right) d x\]
\[ = \frac{1}{2} \int_2^8 (5x + 7) dx\]
\[ = \frac{1}{2} \left[ \frac{5}{2} x^2 + 7x \right]_2^8 \]
\[ = \frac{1}{2}\left[ \frac{5}{2} \times 64 + 56 - \frac{5}{2} \times 4 - 14 \right]\]
\[ = \frac{1}{2} \times 192\]
\[ = 96\text{ sq units }\]
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