Question 1

Given the region bounded above by $$y = x$$ and below by $$y = x^4$ for 0$$ \leq x \leq 1$$, slices perpendicular to the x-axis have cross-sections that are squares. What is the volume of the resulting solid?

Accepted Answer

$$\frac{1}{9}$$

Question 2

For the region bounded by $$y = x^2$ and y = 3x$ from x = 0$ to x = 3$, what is the moment about the x-axis (M_x$) for a lamina of uniform density $$\delta = 1$$?

Accepted Answer

$$\frac{81}{5}$$

Question 3

A solid has a base bounded by $$y = x$$ and $$y = x^4$ for 0$$ \leq x \leq 1$$. If the cross-sections perpendicular to the x-axis are squares, what is the integral expression for the volume?

Accepted Answer

$$\$$int_0$$^1 (x - x^4$)^2 dx$

Question 4

In the second image, what is the y-coordinate of the center of mass for a thin slice between $$y = x^2$ and y = 3x$ at a given x$?

Accepted Answer

$$\frac{3x + x^2$}{2}$$

Question 5

For the region bounded by $$y = x^2$ and y = 3x$ from x = 0$ to x = 3$, what is the integral expression for the moment about the x-axis (M_x$) for a lamina of uniform density $$\delta = 1$$?

Accepted Answer

$$\$$int_0$$^3 \frac{1}{2}(3x + x^2$)(3x - x^2$) dx$$

Question 6

If the area of a square cross-section is $$(x - x^4$)^2$, what is the length of one side of the square?

Accepted Answer

$$x - x^4$

Question 7

For the region bounded by $$y = x^2$ and y = 3x$, what is the area of a thin vertical slice at x$?

Accepted Answer

$$(3x - x^2$) dx$$

Question 8

In the first image, what is the geometric interpretation of the blue squares?

Accepted Answer

They represent the cross-sectional area of the solid formed by perpendicular slices to the x-axis.

Question 9

If the cross-section of a solid is a square with side length $$x - x^4$, what is the volume of the solid from x = 0$ to x = 1$?

Accepted Answer

$$\$$int_0$$^1 (x - x^4$)^2 dx$

Question 10

For the region between $$y = x^2$ and y = 3x$, what is the formula for the area of a thin slice at x$?

Accepted Answer

$$(3x - x^2$) dx$$

Applications of Integrals: Volumes and Moments Using Cross-Sections

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