What are the differences between setting up a volume integral with cross-sections perpendicular to the x-axis vs. the y-axis?

x-axis: Integrate with respect to x, functions in terms of x. y-axis: Integrate with respect to y, functions in terms of y.

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What are the differences between setting up a volume integral with cross-sections perpendicular to the x-axis vs. the y-axis?

x-axis: Integrate with respect to x, functions in terms of x. y-axis: Integrate with respect to y, functions in terms of y.

What are the differences between finding the volume with square vs. rectangular cross-sections?

Square: Need to find the side length 's'. Rectangular: Need to find both width 'w' and height 'h'.

Compare finding the area between two curves and finding the volume with known cross-sections.

Area: Integrate the difference between two functions. Volume: Integrate the area of a cross-section.

What are the differences between disk/washer method and volume with known cross sections?

Disk/Washer: Revolution around an axis, circular cross-sections. Cross Sections: Various shapes, no revolution required.

Compare finding the volume when given a single bounding curve versus two bounding curves.

Single Curve: The axis often acts as the second boundary. Two Curves: Need to find the difference between the functions.

What are the differences between setting up the volume integral when the cross sections are perpendicular to the x-axis vs y-axis?

x-axis: Integrate with respect to x, functions in terms of x. y-axis: Integrate with respect to y, functions in terms of y.

Compare finding the volume with square cross sections and rectangular cross sections.

Square: Need to find the side length 's' and use $A(x) = s^2$ . Rectangular: Need to find both width 'w' and height 'h' and use $A(x) = w * h$ .

Compare finding the area between two curves and finding the volume with known cross sections.

Area: Integrate the difference between two functions, resulting in a two-dimensional area. Volume: Integrate the area of a cross-section, resulting in a three-dimensional volume.

What is the key difference between problems where the cross-sections are perpendicular to the x-axis versus the y-axis?

x-axis: Integrate with respect to x, functions in terms of x. y-axis: Integrate with respect to y, functions in terms of y.

Compare the complexity of finding the volume with square cross sections versus rectangular cross sections.

Square: Involves finding one dimension (side length) and squaring it. Rectangular: Involves finding two dimensions (width and height).

How does the graph of $A(x)$ relate to the volume of the solid?

The area under the curve of $A(x)$ from $a$ to $b$ represents the volume of the solid.

Given the graph of two functions, how can you identify the region that forms the base of the solid?

The region is enclosed between the two curves within the given interval [a, b].

How can you graphically determine the bounds of integration?

Find the x-coordinates (or y-coordinates if integrating with respect to y) of the intersection points of the bounding curves.

If the graph shows cross-sections perpendicular to the y-axis, what does this imply about the integration?

The integration must be performed with respect to y, and the functions must be expressed in terms of y.

How does a steeper slope in the graph of a bounding curve affect the volume of the solid?

A steeper slope can increase the area of the cross-sections, potentially increasing the volume of the solid.

How does the area between two curves on a graph relate to the side length of a square cross-section?

The area between the curves at a given x-value represents the side length 's' of the square cross-section at that x-value.

If the graph of $A(x)$ is always positive, what does this imply about the volume?

The volume will always be positive since we are summing positive areas.

How can you visually estimate the volume of the solid from the graph of $A(x)$ ?

Approximate the area under the curve of $A(x)$ using geometric shapes or numerical methods.

What does it mean if the two curves bounding the region intersect only at one point?

That point defines one of the limits of integration; you may need another boundary to fully define the region.

How can you use a graph to check if you've correctly identified which curve is 'above' or 'to the right'?

Visually confirm that the identified curve is indeed above (for x-axis) or to the right (for y-axis) of the other curve within the integration interval.

Steps to find the volume of a solid with square cross-sections given two bounding curves.

Find the intersection points of the curves (bounds). 2. Determine the side length $s = f(x) - g(x)$ . 3. Square the side length: $A(x) = s^2$ . 4. Integrate: $V = \int_a^b A(x) dx$ .

Steps to find the volume of a solid with rectangular cross-sections perpendicular to the y-axis.

Express bounding curves as functions of y. 2. Find the intersection points (bounds). 3. Determine width and height as functions of y. 4. Find area: $A(y) = w*h$ . 5. Integrate: $V = \int_c^d A(y) dy$ .

How to find the volume of a solid with square cross sections if given $y = x^2$ and $y = 4$ ?

Find bounds: $x = -2, 2$ . 2. $s = 4-x^2$ . 3. $A(x) = (4-x^2)^2$ . 4. $V = \int_{-2}^2 (4-x^2)^2 dx$ .

How to set up the integral for the volume of a solid with rectangular cross sections of height 3, perpendicular to the x-axis, bounded by $y = x$ and $y = x^2$ ?

Find bounds: $x = 0, 1$ . 2. $w = x - x^2$ . 3. $h = 3$ . 4. $V = \int_0^1 3(x - x^2) dx$ .

How to find the volume if the base is bounded by $y=x^3$ , $y=0$ , $x=1$ and the cross sections are squares perpendicular to the x-axis?

Find bounds: $x = 0, 1$ . 2. $s = x^3 - 0 = x^3$ . 3. $A(x) = (x^3)^2 = x^6$ . 4. $V = \int_0^1 x^6 dx$ .

How to find the volume if the base is bounded by $x=y^2$ , $x=4$ and the cross sections are rectangles with height $y$ perpendicular to the y-axis?

Find bounds: $y = -2, 2$ . 2. $w = 4 - y^2$ . 3. $h = y$ . 4. $V = \int_{-2}^2 y(4 - y^2) dy$ .

How do you determine the limits of integration when the region is bounded by $y = x^2$ and $y = sqrt{x}$ ?

Set $x^2 = \sqrt{x}$ and solve for $x$ to find the intersection points, which are the limits of integration.

What is the general strategy for solving volume problems with known cross-sections?

Visualize the solid. 2. Determine the shape and area of the cross-section. 3. Find the limits of integration. 4. Set up and evaluate the integral.

If the cross-sections are perpendicular to the y-axis, how do you express the bounding curves?

Express the curves as functions of y, i.e., $x = f(y)$ and $x = g(y)$ .

How do you handle a problem where the height of the rectangular cross-section is given as a function of x?

Include the height function in the area function $A(x) = w(x) cdot h(x)$ and integrate with respect to x.