What would you like to understand?

The antiderivative of $\sqrt{u}$ is found by rewriting the square root as a fractional power, $u^{1/2}$, and then applying the power rule for integration. The power rule states that $\int u^n du = \frac{u^{n+1}}{n+1}$.

Conditions: The integrand is $\sqrt{u}$, which is equivalent to $u^{1/2}$.; The power rule for integration is applicable.; $u \ge 0$ for the real-valued square root form.

The antiderivative of $\sqrt{u}$ is found by rewriting the square root as a fractional power, $u^{1/2}$, and then applying the power rule for integration. The power rule states that $\int u^n du = \frac{u^{n+1}}{n+1}$.

Conditions: The integrand is $\sqrt{u}$, which is equivalent to $u^{1/2}$.; The power rule for integration is applicable.; $u \ge 0$ for the real-valued square root form.

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 $n = 1/2$.

Factoring and cancelling resolves the limit by simplifying the expression to a form that is defined near the target point. For $x \ne 2$ near 2, the expression $\frac{x^2+3x-10}{x^2-2x}$ factors into $\frac{(x-2)(x+5)}{x(x-2)}$.

Conditions: The limit is as $x \to 2$.; The original expression is undefined at $x=2$ (indeterminate form $0/0$).; Cancellation is valid only for $x \ne 2$.

Factoring and cancelling resolves the limit by simplifying the expression to a form that is defined near the target point. For $x \ne 2$ near 2, the expression $\frac{x^2+3x-10}{x^2-2x}$ factors into $\frac{(x-2)(x+5)}{x(x-2)}$.

Conditions: The limit is as $x \to 2$.; The original expression is undefined at $x=2$ (indeterminate form $0/0$).; Cancellation is valid only for $x \ne 2$.

The presenter factors out the common coefficient $\frac{2}{3}$ to make the subsequent arithmetic simplification easier. After evaluating the antiderivative at the upper and lower bounds, both terms inside the brackets contain the factor $\frac{2}{3}$.

Conditions: The expression to evaluate is $\frac{4}{9}\left[\frac{2}{3}\cdot 9^{3/2}-\frac{2}{3}\cdot 1^{3/2}\right]$.; Both bracketed terms contain the same factor $\frac{2}{3}$.

The presenter factors out the common coefficient $\frac{2}{3}$ to make the subsequent arithmetic simplification easier. After evaluating the antiderivative at the upper and lower bounds, both terms inside the brackets contain the factor $\frac{2}{3}$.

Conditions: The expression to evaluate is $\frac{4}{9}\left[\frac{2}{3}\cdot 9^{3/2}-\frac{2}{3}\cdot 1^{3/2}\right]$.; Both bracketed terms contain the same factor $\frac{2}{3}$.

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.

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 $\frac{2}{3}x^{3/2}$.

Clearing the inner fraction resolves the limit by simplifying the complex rational expression into a polynomial form. Near $x=-1$, with $x\ne-1$, multiplying the numerator and denominator of $\frac{3+3x}{1/x+1}$ by $x$ gives $\frac{3x+3x^2}{1+x}$.

Conditions: The limit is as $x \to -1$.; The expression contains an inner fraction $1/x$ in the denominator.; Multiplication by $x$ is valid for $x \ne 0$, and specifically near $x=-1$ where $x \ne 0$.; The original expression is undefined at $x=-1$ (indeterminate form $0/0$).

Clearing the inner fraction resolves the limit by simplifying the complex rational expression into a polynomial form. Near $x=-1$, with $x\ne-1$, multiplying the numerator and denominator of $\frac{3+3x}{1/x+1}$ by $x$ gives $\frac{3x+3x^2}{1+x}$.

Conditions: The limit is as $x \to -1$.; The expression contains an inner fraction $1/x$ in the denominator.; Multiplication by $x$ is valid for $x \ne 0$, and specifically near $x=-1$ where $x \ne 0$.; The original expression is undefined at $x=-1$ (indeterminate form $0/0$).

The substitution $u = 1 + \frac{9}{4}x$ is chosen because it is exactly the expression under the radical in the arc-length integral $\int_0^{32/9}\sqrt{1+\frac{9}{4}x}\,dx$. By setting $u$ to this inner expression, the complicated integrand simplifies to $\sqrt{u}$, which is much easier to integrate using the power rule.

Conditions: The integral to evaluate is $\int_0^{32/9}\sqrt{1+\frac{9}{4}x}\,dx$.; The integrand contains a composite expression under a square root.

The substitution $u = 1 + \frac{9}{4}x$ is chosen because it is exactly the expression under the radical in the arc-length integral $\int_0^{32/9}\sqrt{1+\frac{9}{4}x}\,dx$. By setting $u$ to this inner expression, the complicated integrand simplifies to $\sqrt{u}$, which is much easier to integrate using the power rule.

Conditions: The integral to evaluate is $\int_0^{32/9}\sqrt{1+\frac{9}{4}x}\,dx$.; The integrand contains a composite expression under a square root.

To evaluate $\int_2^3 \sqrt{x} dx$, first find the antiderivative $\frac{2}{3}x^{3/2}$. Then, apply the Fundamental Theorem of Calculus by subtracting the value of the antiderivative at the lower bound from its value at the upper bound: $\left[\frac{2}{3}x^{3/2}\right]_2^3 = \frac{2}{3}(3)^{3/2} - \frac{2}{3}(2)^{3/2}$.

Conditions: The integral is $\int_2^3 \sqrt{x} dx$.; The antiderivative is $\frac{2}{3}x^{3/2}$.; The bounds are 2 and 3.

To evaluate $\int_2^3 \sqrt{x} dx$, first find the antiderivative $\frac{2}{3}x^{3/2}$. Then, apply the Fundamental Theorem of Calculus by subtracting the value of the antiderivative at the lower bound from its value at the upper bound: $\left[\frac{2}{3}x^{3/2}\right]_2^3 = \frac{2}{3}(3)^{3/2} - \frac{2}{3}(2)^{3/2}$.

Conditions: The integral is $\int_2^3 \sqrt{x} dx$.; The antiderivative is $\frac{2}{3}x^{3/2}$.; The bounds are 2 and 3.