# The base of a certain solid is the triangle with vertices at (-8,4), (4,4), and the origin. Cross-sections perpendicular to the y-axis are squares. How do you find the volume of the solid?

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#### Explanation

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#### Explanation:

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Jim H Share
Dec 15, 2016

#### Explanation:

Here is a picture (graph) of the base (in blue) with a thin slice taken perpendicular to the $y$-axis (in black).

The thinkness of this representative slice is $\mathrm{dy}$ so we need the sides of the square to be expressed in terms of $y$ (not $x$).

The line on the right contains $\left(0 , 0\right)$ and $\left(4 , 4\right)$. Its equation is $y = x$ or $x = y$

The line on the left contains $\left(0 , 0\right)$ and $\left(- 8 , 4\right)$. Its equation is $y = - \frac{1}{2} x$ or $x = - 2 y$

The side of the square built on the representative slice is $s = {x}_{\text{greater"-x_"lesser" = x_"on the right"-x_"on the left}}$.

So, $s = y - \left(- 2 y\right) = 3 y$.

The volume of the representative slice is

${s}^{2} \cdot \text{thickness} = {\left(3 y\right)}^{2} \mathrm{dy}$.

The values of $y$ vary from $0$ to $4$, so the volume of the solid is

$V = {\int}_{0}^{4} {\left(3 y\right)}^{2} \mathrm{dy} = {\left.3 {y}^{3}\right]}_{0}^{4} = 192$ (cubic units)

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