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The antiderivative of $\sqrt{x}$ is $\frac{2}{3}x^{3/2}$, derived by rewriting the function as $x^{1/2}$ and applying the power rule for integration. This result is specifically used to evaluate definite integrals after transforming the bounds, where the constant of integration is omitted.

Conditions: The function is $\sqrt{x}$.; The power rule for integration is applied.; The exponent $n = 1/2$.

The antiderivative of $\sqrt{x}$ is $\frac{2}{3}x^{3/2}$, derived by rewriting the function as $x^{1/2}$ and applying the power rule for integration. This result is specifically used to evaluate definite integrals after transforming the bounds, where the constant of integration is omitted.

Conditions: The function is $\sqrt{x}$.; The power rule for integration is applied.; The exponent $n = 1/2$.

The definite integral $\int_0^{\ln 2} e^u \sqrt{1+e^u} du$ evaluates to the exact value $2\sqrt{3} - \frac{4\sqrt{2}}{3}$. This result is obtained by applying the substitution $x = 1 + e^u$, which transforms the bounds to 2 and 3, and then evaluating the antiderivative $\frac{2}{3}x^{3/2}$ at these new limits.

Conditions: The integral is $\int_0^{\ln 2} e^u \sqrt{1+e^u} du$.; The substitution $x = 1 + e^u$ is used.; The bounds are converted to 2 and 3.; The antiderivative is $\frac{2}{3}x^{3/2}$.

The definite integral $\int_0^{\ln 2} e^u \sqrt{1+e^u} du$ evaluates to the exact value $2\sqrt{3} - \frac{4\sqrt{2}}{3}$. This result is obtained by applying the substitution $x = 1 + e^u$, which transforms the bounds to 2 and 3, and then evaluating the antiderivative $\frac{2}{3}x^{3/2}$ at these new limits.

Conditions: The integral is $\int_0^{\ln 2} e^u \sqrt{1+e^u} du$.; The substitution $x = 1 + e^u$ is used.; The bounds are converted to 2 and 3.; The antiderivative is $\frac{2}{3}x^{3/2}$.

The real domain of the function $y=\sqrt{9-x^2}$ is the closed interval $[-3, 3]$. This restriction exists because, over the real numbers, the expression inside a square root (the radicand) must be nonnegative for the principal square root to be real-valued.

Conditions: Real-valued interpretation of the square root.; Using the principal square root convention.; The radicand is $9 - x^2$.

The real domain of the function $y=\sqrt{9-x^2}$ is the closed interval $[-3, 3]$. This restriction exists because, over the real numbers, the expression inside a square root (the radicand) must be nonnegative for the principal square root to be real-valued.

Conditions: Real-valued interpretation of the square root.; Using the principal square root convention.; The radicand is $9 - x^2$.

The final numerical value of the area moment of inertia $I_y$ for the specific shaded region about the centroidal y-axis is approximately $0.762 \text{ m}^4$. This value is obtained by evaluating the definite integral $\int_0^2 x^2 (2 - \sqrt{2} x^{1/2}) dx$, which simplifies to $\frac{2^4}{3} - \sqrt{2} \left(\frac{2}{7}\right) 2^{3.5}$.

Conditions: The region is bounded by $y = 2$ (top) and $y = \sqrt{2} x^{1/2}$ (bottom) from $x=0$ to $x=2$.; The axis is the centroidal y-axis.; Units are in meters.

The final numerical value of the area moment of inertia $I_y$ for the specific shaded region about the centroidal y-axis is approximately $0.762 \text{ m}^4$. This value is obtained by evaluating the definite integral $\int_0^2 x^2 (2 - \sqrt{2} x^{1/2}) dx$, which simplifies to $\frac{2^4}{3} - \sqrt{2} \left(\frac{2}{7}\right) 2^{3.5}$.

Conditions: The region is bounded by $y = 2$ (top) and $y = \sqrt{2} x^{1/2}$ (bottom) from $x=0$ to $x=2$.; The axis is the centroidal y-axis.; Units are in meters.