The area of the region bounded by the curve y | vedclass

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Question

(A) 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} |f(

Vedclass · Accepted Answer

$18$

Vedclass · Answer

(A) 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 $A = \int_{a}^{b} |f(x)| dx$.Here,the curve is $y = 9 - x^2$,and the limits are $x=0$ to $x=3$.Since $9 - x^2 \geq 0$ for $x \in [0, 3]$,the area is given by:$A = \int_{0}^{3} (9 - x^2) dx$$A = [9x - \frac{x^3}{3}]_{0}^{3}$$A = (9(3) - \frac{3^3}{3}) - (9(0) - \frac{0^3}{3})$$A = (27 - \frac{27}{3}) - 0$$A = 27 - 9 = 18$ sq. units.Therefore,the correct option is $A$.

The area of the region bounded by the curve $y=9-x^2$,the $X$-axis,and the lines $x=0$ and $x=3$ is . . . . . . sq. units.

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