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The antiderivative of $\sqrt{x}$ is $\frac{2}{3}x^{3/2}$. This is found by rewriting $\sqrt{x}$ as $x^{1/2}$ and applying the power rule for integration, which states that $\int x^n dx = \frac{x^{n+1}}{n+1}$ for $n \neq -1$.

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

The antiderivative of $\sqrt{x}$ is $\frac{2}{3}x^{3/2}$. This is found by rewriting $\sqrt{x}$ as $x^{1/2}$ and applying the power rule for integration, which states that $\int x^n dx = \frac{x^{n+1}}{n+1}$ for $n \neq -1$.

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

The exact value of the definite integral $\int_{-3}^{3} \sqrt{9-x^2}\,dx$ is $\frac{9\pi}{2}$. This is derived by interpreting the integral as the geometric area of the upper semicircle of a circle with radius 3.

Conditions: Interpret the integral as area under the graph on $[-3,3]$.; Recognize the graph as the upper semicircle of radius 3.; Exact equality for the displayed definite integral.

The exact value of the definite integral $\int_{-3}^{3} \sqrt{9-x^2}\,dx$ is $\frac{9\pi}{2}$. This is derived by interpreting the integral as the geometric area of the upper semicircle of a circle with radius 3.

Conditions: Interpret the integral as area under the graph on $[-3,3]$.; Recognize the graph as the upper semicircle of radius 3.; Exact equality for the displayed definite integral.

The final value is $2\sqrt{3} - \frac{4\sqrt{2}}{3}$. This is obtained by evaluating the antiderivative $\frac{2}{3}x^{3/2}$ at the converted bounds $x=3$ and $x=2$ and subtracting the results.

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 final value is $2\sqrt{3} - \frac{4\sqrt{2}}{3}$. This is obtained by evaluating the antiderivative $\frac{2}{3}x^{3/2}$ at the converted bounds $x=3$ and $x=2$ and subtracting the results.

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.

For the arc length formula $\int_{a}^{b} \sqrt{1 + (f'(x))^2} dx$ to be valid, the function $f(x)$ must be continuous on the closed interval $[a, b]$, and its derivative $f'(x)$ must also be continuous on $[a, b]$. These conditions ensure that the curve is smooth enough for the integral to accurately represent its length.

Conditions: The curve is defined by a function $y = f(x)$.; The interval of integration is $[a, b]$.; $f(x)$ is continuous on $[a, b]$.; $f'(x)$ is continuous on $[a, b]$.

For the arc length formula $\int_{a}^{b} \sqrt{1 + (f'(x))^2} dx$ to be valid, the function $f(x)$ must be continuous on the closed interval $[a, b]$, and its derivative $f'(x)$ must also be continuous on $[a, b]$. These conditions ensure that the curve is smooth enough for the integral to accurately represent its length.

Conditions: The curve is defined by a function $y = f(x)$.; The interval of integration is $[a, b]$.; $f(x)$ is continuous on $[a, b]$.; $f'(x)$ is continuous on $[a, b]$.

The curly d symbol ($\partial$) represents a partial derivative. It means taking the derivative of a multivariable function with respect to one specific variable at a time, while treating all other variables as constants.

Conditions: The function has more than one independent variable.; All variables except the differentiation variable are treated as constants during the partial derivative.

The curly d symbol ($\partial$) represents a partial derivative. It means taking the derivative of a multivariable function with respect to one specific variable at a time, while treating all other variables as constants.

Conditions: The function has more than one independent variable.; All variables except the differentiation variable are treated as constants during the partial derivative.