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(A) The area of the region bounded by the curve $y^{2}=x$,the lines $x=1$ and $x=4$,and the $x$-axis in the first quadrant is given by the integral of

Vedclass · Accepted Answer

$\frac{14}{3}$ square units

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(A) The area of the region bounded by the curve $y^{2}=x$,the lines $x=1$ and $x=4$,and the $x$-axis in the first quadrant is given by the integral of $y$ with respect to $x$ from $x=1$ to $x=4$.Since $y^{2}=x$ and we are in the first quadrant,$y = \sqrt{x}$.Area $= \int_{1}^{4} y \, dx = \int_{1}^{4} \sqrt{x} \, dx$$= \left[ \frac{x^{\frac{3}{2}}}{\frac{3}{2}} ight]_1^4$$= \frac{2}{3} \left[ x^{\frac{3}{2}} ight]_1^4$$= \frac{2}{3} \left( 4^{\frac{3}{2}} - 1^{\frac{3}{2}} ight)$$= \frac{2}{3} (8 - 1)$$= \frac{2}{3} 	imes 7 = \frac{14}{3}$ square units.

Find the area of the region bounded by the curve $y^{2}=x$,the lines $x=1$,$x=4$,and the $x$-axis in the first quadrant.

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