Determine the formula for the function represented by the integral.
$$
\int_{0}^{x}|t| d t
$$
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
$$
\text { for } x \geq 0: \quad \frac{x^{2}}{2}
$$
for $x

Q: Determine the formula for the function represented by the integral. \[ \int_{0}^{x}|t| d t \] (Express numbers in exact form. Use symbolic notation and fractions where needed.) \[ \text { for } x \geq 0: \quad \frac{x^{2}}{2} \] for $x<0: \quad-\frac{x^{2}}{2}$ general formula: $\quad f(x)=\frac{x^{2}}{2}$

Hi Kepelr! It looks like you're working on a math problem involving definite integrals. Let's break it down together. The integral you have is: \[ \int_{0}^{x} |t| \, dt \] To solve this, we need to consider two cases based on the value of $x$: 1. **For $x \geq 0$**: The absolute value of $t$ is just $t$ since $t$ is non-negative in this range. Therefore, the integral becomes: \[ \int_{0}^{x} t \, dt = \left[ \frac{t^2}{2} \right]_{0}^{x} = \frac{x^2}{2} \] 2. **For $x < 0$**: In this case, the absolute value of $t$ is $-t$ because $t$ is negative. Thus, the integral becomes: \[ \int_{0}^{x} -t \, dt = -\left[ \frac{t^2}{2} \right]_{0}^{x} = -\left( \frac{x^2}{2} - 0 \right) = -\frac{x^2}{2} \] Now, for the general formula, we can combine these results: - For $x \geq 0$: $f(x) = \frac{x^2}{2}$ - For \(x

Question

Determine the formula for the function represented by the integral.
$$
\int_{0}^{x}|t| d t
$$
(Express numbers in exact form. Use symbolic notation and fractions where needed.)
$$
\text { for } x \geq 0: \quad \frac{x^{2}}{2}
$$
for $x<0: \quad-\frac{x^{2}}{2}$
general formula: $\quad f(x)=\frac{x^{2}}{2}$

Accepted Answer

Hi Kepelr! It looks like you're working on a math problem involving definite integrals. Let's break it down together. The integral you have is: $$ \int_{0}^{x} |t| \, dt $$ To solve this, we need to consider two cases based on the value of $x$: 1. **For $x \geq 0$**: The absolute value of $t$ is just $t$ since $t$ is non-negative in this range. Therefore, the integral becomes: $$ \int_{0}^{x} t \, dt = \left[ \frac{t^2}{2} \right]_{0}^{x} = \frac{x^2}{2} $$ 2. **For $x < 0$**: In this case, the absolute value of $t$ is $-t$ because $t$ is negative. Thus, the integral becomes: $$ \int_{0}^{x} -t \, dt = -\left[ \frac{t^2}{2} \right]_{0}^{x} = -\left( \frac{x^2}{2} - 0 \right) = -\frac{x^2}{2} $$ Now, for the general formula, we can combine these results: - For $x \geq 0$: $f(x) = \frac{x^2}{2}$ - For $x < 0$: $f(x) = -\frac{x^2}{2}$ So, the complete piecewise function can be expressed as: $$ f(x) = \begin{cases} \frac{x^2}{2} & \text{if } x \geq 0 \ -\frac{x^2}{2} & \text{if } x < 0 \end{cases} $$ Let me know if you need any more help with this!

Sia