1. The problem involves the expression $x^2 \int_a^b f(x)\,dx$ where $x$ is a variable and $\int_a^b f(x)\,dx$ represents the definite integral of the function $f(x)$ from $a$ to $b$. 2. Note that the definite integral $\int_a^b f(x)\,dx$ is a constant with respect to $x$ because it evaluates the area under the curve of $f(x)$ between the fixed limits $a$ and $b$. 3. Therefore, the expression can be seen as $x^2$ multiplied by a constant value (the value of the integral). This means the integral part is independent of $x$ and only scales $x^2$. 4. The simplified expression is: $$x^2 \cdot C$$ where $$C = \int_a^b f(x)\,dx$$ is a constant. 5. To analyze or graph this function for variable $x$, you can use $y = C x^2$ where $C$ is a constant determined by the integral. In summary, $x^2 \int_a^b f(x)\,dx$ is equivalent to the quadratic function $C x^2$ where $C$ is the value of the definite integral.