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) $\int_{0}^{p}[f(x)-g(x)] \cdot x d x$

Question

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) $\int_{0}^{p}[f(x)-g(x)] \cdot x d x$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To find the volume of the solid whose base is the region $R$ enclosed by the graphs of $f(x)$ and $g(x)$ and the line $x=p$, and whose cross sections perpendicular to the $x$-axis are squares, we need to use the formula for the volume of a solid with known cross-sectional area. ‖ ‖ 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.⚹◊What is the formula for calculating the volume of a solid formed by the region enclosed by two functions using definite integrals?⍭ Generate me a similar question◊

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