Consider the region bounded by the x-axis, the line x = 1, the line x = 4, and the curve $y=x^{2}$ .Imagine that this region is the base of a solid.Each cross-section of this solid perpendicular to the x-axis is a triangle of height H . The volume of this solid can be found exactly and can be written as $\frac{H B}{6}$ .What is B?
A) $63$
B) $15$
C) $3$
D) $64$

Question

Quizplus · Accepted Answer

The volume of the solid with triangular cross-sections perpendicular to the x-axis can be calculated using the formula for the volume of a triangular prism, $V = \frac{1}{2}bhH$, where $b$ is the base of the triangle (in this case, the distance between the curve $y=x^2$ and the x-axis), $h$ is the height of the triangle (given as $H$), and $H$ is the depth of the prism (in this case, the differential element $dx$). Integrating from $x=1$ to $x=4$, we get the volume as $V = \int_{1}^{4} \frac{1}{2} x^2 H dx$. Solving this integral, we get $V = \frac{H}{2} \left[\frac{x^3}{3}\right]_{1}^{4} = \frac{H}{2} \left(\frac{64}{3} - \frac{1}{3}\right) = \frac{H}{2} \cdot \frac{63}{3} = \frac{H}{2} \cdot 21 = \frac{H \cdot 63}{6}$. Therefore, $B = 63$.

Consider the Region Bounded by the X-Axis, the Line X y=x2y=x^{2}y=x2

Consider the Region Bounded by the X-Axis, the Line X $y=x^{2}$