The area (in sq. units) of the region bounded | vedclass

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(D) The area of the region bounded by the curves $y = f(x)$ and $y = g(x)$ between $x = a$ and $x = b$ is given by $\int_{a}^{b} |f(x) - g(x)| dx$.Her

Vedclass · Accepted Answer

$\frac{15}{2}$

Vedclass · Answer

(D) The area of the region bounded by the curves $y = f(x)$ and $y = g(x)$ between $x = a$ and $x = b$ is given by $\int_{a}^{b} |f(x) - g(x)| dx$.Here,$f(x) = x^2 + 2$ and $g(x) = x + 1$. For $x \in [0, 3]$,$x^2 + 2 \geq x + 1$ because $x^2 - x + 1 = (x - 0.5)^2 + 0.75 > 0$.Therefore,the required area is $\int_{0}^{3} ((x^2 + 2) - (x + 1)) dx$.$= \int_{0}^{3} (x^2 - x + 1) dx$$= \left[ \frac{x^3}{3} - \frac{x^2}{2} + x \right]_{0}^{3}$$= \left( \frac{3^3}{3} - \frac{3^2}{2} + 3 \right) - (0)$$= \left( 9 - 4.5 + 3 \right) = 7.5 = \frac{15}{2}$ sq. units.

The area (in sq. units) of the region bounded by the parabola $y = x^2 + 2$ and the lines $y = x + 1$,$x = 0$,and $x = 3$ is:

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