Question

Math

Posted 4 months ago

```
Let $R$ be the region enclosed by the graphs of functions $f$ and $g$ and the line $x=p$.
Region $R$ is the base of a solid whose cross sections perpendicular to the $x$ -axis are squares.
Which one of the definite integrals gives the volume of the solid?
Choose 1 answer:
(A) $\int_{q}^{r}[f(x)-g(x)] \cdot x d x$
(B) $\int_{0}^{p}[f(x)-g(x)]^{2} d x$
(C) $\int_{q}^{r}[f(x)-g(x)]^{2} d x$
(D) $\int_{0}^{p}[f(x)-g(x)] \cdot x d x$
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 2

The area of each square cross-section is given by the square of the difference between the functions $f(x)$ and $g(x)$, i.e., $[f(x) - g(x)]^2$

step 3

The volume of the solid is then the integral of this area from $x=0$ to $x=p$. Therefore, the definite integral that gives the volume of the solid is $\int_{0}^{p} [f(x) - g(x)]^2 \, dx$

B

Key Concept

Volume of a solid with square cross-sections

Explanation

The volume of a solid with square cross-sections perpendicular to the $x$-axis is found by integrating the square of the difference between the functions over the given interval.

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