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Question
Find the area of the region bounded by the following curves, the X-axis, and the given lines:
y = `sqrt(6x + 4), x = 0, x = 2`
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Solution
Let A be the required area.
∴ A = `int_0^2 y dx`
A = `int_0^2 sqrt(6x + 4) dx`
= `int_0^2 (6x + 4)^(1/2) dx`
A = `[(6x + 4)^{1/2 + 1}/((1/2 + 1) xx 6)]_0^2`
= `[(6x + 4)^{3/2}/(3/2 xx 6)]_0^2`
= `[((6x + 4)^{3/2})/9]_0^2`
= `1/9[(6x + 4)^{3/2}]_0^2`
= `1/9[(6 xx 2 + 4)^{3/2} - (6 xx 0 + 4)^{3/2}]`
= `1/9[(12 + 4)^{3/2} - (4)^{3/2}]`
= `1/9[(16)^{3/2} - (4)^{3/2}]`
= `1/9[(4^2)^{3/2} - (2^2)^{3/2}]`
∴ A = `1/9[4^3 - 2^3]`
= `1/9(64 - 8)`
= `1/9 xx 56`
= `56/9` sq.units.
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