The area of the region bounded by the parabol | vedclass

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(C) The area $A$ of the region bounded by the curve $y = f(x)$,the $X$-axis,and the lines $x = a$ and $x = b$ is given by the integral $A = \int_{a}^{

Vedclass · Accepted Answer

$\frac{13}{3}$

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(C) The area $A$ of the region bounded by the curve $y = f(x)$,the $X$-axis,and the lines $x = a$ and $x = b$ is given by the integral $A = \int_{a}^{b} y \, dx$. Given the curve $y = x^2 + 2$,the limits $x = 1$ and $x = 2$,the area is: $A = \int_{1}^{2} (x^2 + 2) \, dx$ Integrating the function: $A = [\frac{x^3}{3} + 2x]_{1}^{2}$ Substituting the upper and lower limits: $A = (\frac{2^3}{3} + 2(2)) - (\frac{1^3}{3} + 2(1))$ $A = (\frac{8}{3} + 4) - (\frac{1}{3} + 2)$ $A = (\frac{8 + 12}{3}) - (\frac{1 + 6}{3})$ $A = \frac{20}{3} - \frac{7}{3} = \frac{13}{3}$ Thus,the area is $\frac{13}{3}$ sq. units.

The area of the region bounded by the parabola $y = x^2 + 2$,the $X$-axis,and the lines $x = 1$ and $x = 2$ is . . . . . . sq. units.

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