Volumes (Cross Sections Method)

Volumes - Cross Sections Method

The volume of a solid with parallel cross sections that are perpendicular to the base is given by

$\displaystyle V=\int_{x=a}^{x=b}A\left(x\right)\ dx\ \ \text{or}\ \ \int_{y=c}^{y=d}A\left(y\right)\ dy$

where $A\left(x\right)\ \ \text{or}\ \ A\left(y\right)$ are functions representing the areas of these cross sections.

Example

Write an integral that represents the volume of a cube of side length $L$

Since the cross sections that are perpendicular to the base of the cube are all squares that have the same size, the cross sectional area is just $L^2$

Let's suppose that one end of this cube starts at $x=0$ and goes to $x=L$ . Then, using our formula, we know that the volume is

$\displaystyle V=\int_0^LL^2dx=\left[L^2x\right]_0^L=L^3$

Practice: Volume Using Cross Sections

Suppose that a solid has a circular base of radius $a$ . The parallel cross sections that are perpendicular to the base are isosceles triangles where the length of the two identical sides are twice the length of the base of the triangle. Find the integral representing the volume of this solid.

$\displaystyle V=\int_{-a}^a(a^2-x^2)dx$

$\displaystyle V=\int_{-a}^a\frac{a^2-x^2}{2}dx$

$\displaystyle V=\int_{-a}^a\sqrt{15}\left(a^2-x^2\right)dx$

$\displaystyle V=\int_{-a}^ay^2\ dy$

$\displaystyle V=\int_{-a}^a\sqrt{3}y^2\ dy$

I don't know

Practice: Volumes Using Cross Sections

Find the volume of a cone whose base is a circle with radius $R$ and whose height is $h$ .

Hint: R and h are constants, the only variable you should use is $x$ .

$\displaystyle \frac{\pi R^2h^2}{2}$

$\displaystyle \pi R^2h$

$\displaystyle \frac{\pi R^2h}{3}$

$\displaystyle \frac{2\pi R^2h}{3}$

$\displaystyle \frac{3\pi R^2h}{2}$

I don't know

Extra Practice

Volumes of Revolution by Cross Sections

Find the volume of the solid obtained by rotating the region bounded by the curves $y=5-x^2$ and $y=4$ about the line 𝑦 = 2.

Volumes of Revolution by Cross Sections

Evaluate the volume obtained by rotating the region enclosed by the curves $y=e^x,\ \ y=1,\ \ x=e$ around the x-axis

Volumes of Revolution by Cross Sections

Find the volume of the solid obtained by rotating the region bounded by the curves $y=5-x^2$ and $y=4$ about the line 𝑦 = 2.

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