The area bounded by the curve y = 2x^2,the X- | vedclass

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Question

(A) The area $A$ 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} |f(x)| \, d

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$\frac{2}{3}$

Vedclass · Answer

(A) The area $A$ bounded by the curve $y = f(x)$,the $X$-axis,and the lines $x = a$ and $x = b$ is given by the integral $A = \int_{a}^{b} |f(x)| \, dx$. Given the curve $y = 2x^2$,the $X$-axis,and the line $x = 1$,the region is bounded from $x = 0$ to $x = 1$. Thus,the area $A = \int_{0}^{1} 2x^2 \, dx$. Evaluating the integral: $A = 2 \left[ \frac{x^3}{3} \right]_{0}^{1}$. $A = 2 \left( \frac{1^3}{3} - \frac{0^3}{3} \right) = 2 \left( \frac{1}{3} \right) = \frac{2}{3}$ sq. units. Therefore,the correct option is $A$.

The area bounded by the curve $y = 2x^2$,the $X$-axis,and the line $x = 1$ is . . . . . . sq. units.

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