Find the volume of the colid of \( y=2 x+1, y=0, z=1 \) a A \( \pi\left(\frac{11}{3}+7\right) \) units \( ^{3} \) B. None of the options C \( \pi\left(\frac{23}{3}+7\right) \) units \( ^{3} \) D \( \pi\left(\frac{2 \pi}{2}-1\right) \) units E \( -\pi\left(\frac{23}{2}-7\right) \) units \( { }^{3} \)

To find the volume of the solid defined by the equations \( y = 2x + 1 \), \( y = 0 \), and \( z = 1 \), we first need to understand the region in the \(xy\)-plane that is bounded by these lines. 1. **Finding the intersection points**: - The line \( y = 2x + 1 \) intersects the line \( y = 0 \) when: \[ 0 = 2x + 1 \implies 2x = -1 \implies x = -\frac{1}{2} \] - Thus, the intersection point is \( \left(-\frac{1}{2}, 0\right) \). 2. **Finding the volume**: - The region in the \(xy\)-plane is bounded by \( y = 2x + 1 \) and \( y = 0 \) from \( x = -\frac{1}{2} \) to the point where \( y = 0 \) intersects the line \( y = 2x + 1 \). - The line \( y = 2x + 1 \) intersects the \(y\)-axis at \( (0, 1) \). 3. **Setting up the volume integral**: - The volume of the solid can be found by integrating the area of the cross-section (which is a rectangle with height \( z = 1 \)) over the region in the \(xy\)-plane. - The area \( A \) of the cross-section at a given \( x \) is given by: \[ A = \text{height} \times \text{width} = 1 \times (2x + 1) = 2x + 1 \] - The volume \( V \) is then given by: \[ V = \int_{-\frac{1}{2}}^{0} (2x + 1) \, dx \] 4. **Calculating the integral**: \[ V = \int_{-\frac{1}{2}}^{0} (2x + 1) \, dx = \left[ x^2 + x \right]_{-\frac{1}{2}}^{0} \] - Evaluating at the bounds: \[ = \left[ 0^2 + 0 \right] - \left[ \left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right) \right] \] \[ = 0 - \left[ \frac{1}{4} - \frac{1}{2} \right] = 0 - \left[ \frac{1}{4} - \frac{2}{4} \right] = 0 - \left[ -\frac{1}{4} \right] = \frac{1}{4} \] 5. **Finding the volume**: - Since the height of the solid is \( z = 1 \), the volume is: \[ V = 1 \times \frac{1}{4} = \frac{1}{4} \] 6. **Final volume**: - The volume of the solid is \( \frac{1}{4} \) cubic units. Since none of the options provided match this volume, the answer is: **B. None of the options**.