Define 'cross-section' in the context of volumes.

A two-dimensional shape formed by slicing a three-dimensional solid.

Flip to see [answer/question]

Revise later

SpaceTo flip

If confident

All Flashcards

Define 'cross-section' in the context of volumes.

A two-dimensional shape formed by slicing a three-dimensional solid.

What is $A(x)$ in the volume formula?

Area of the cross-section perpendicular to the x-axis at a given x.

Define the term 'solid of revolution'.

A 3D solid formed by rotating a 2D shape around an axis.

What does ' $dx$ ' represent in the volume integral?

Infinitesimally small thickness of the cross-section.

What is the significance of the interval $[a, b]$ ?

The limits of integration, defining the region's boundaries.

Define the term 'volume'.

The amount of three-dimensional space occupied by an object.

What is the formula for the area of a square?

Area of a square is $s^2$ , where s is the side length.

What is the formula for the area of a rectangle?

Area of a rectangle is $w*h$ , where w is the width and h is the height.

What is the meaning of 'perpendicular'?

Intersecting at or forming right angles (90 degrees).

Define 'definite integral'.

Integral evaluated between specific upper and lower limits, resulting in a numerical value.

Volume of a solid with known cross-sections.

$V = \int_a^b A(x) dx$

Volume of a solid with square cross-sections.

$V = \int_a^b s^2 dx$

Volume of a solid with rectangular cross-sections.

$V = \int_a^b w cdot h dx$

Area of a square.

$A = s^2$

Area of a rectangle.

$A = w cdot h$

How do you find the side length 's' of a square cross section when given two bounding curves f(x) and g(x), where f(x) is above g(x)?

$s = f(x) - g(x)$

How do you find the width 'w' of a rectangular cross section perpendicular to the y-axis when given two bounding curves f(y) and g(y), where f(y) is to the right of g(y)?

$w = f(y) - g(y)$

How to find the intersection points of two curves, $f(x)$ and $g(x)$ ?

Set $f(x) = g(x)$ and solve for $x$ .

If integrating with respect to 'y', what does the volume formula become for general cross sections?

$V = \int_c^d A(y) dy$

How do you express $y = x^3$ as a function of y?

$x = \sqrt[3]{y}$

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).