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Steps to find the volume of a solid revolved around the x-axis using the disc method?

Identify $f(x)$ and bounds $a$ and $b$ . 2. Set up the integral: $V = \int_{a}^{b} \pi [f(x)]^2 dx$ . 3. Evaluate the integral.

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Steps to find the volume of a solid revolved around the x-axis using the disc method?

Identify $f(x)$ and bounds $a$ and $b$ . 2. Set up the integral: $V = \int_{a}^{b} \pi [f(x)]^2 dx$ . 3. Evaluate the integral.

Steps to find the volume of a solid revolved around the y-axis using the disc method?

Express $x$ as $f(y)$ and identify bounds $c$ and $d$ . 2. Set up the integral: $V = \int_{c}^{d} \pi [f(y)]^2 dy$ . 3. Evaluate the integral.

How do you determine the limits of integration when revolving around the x-axis?

Find the x-values where the region begins and ends along the x-axis.

How do you determine the limits of integration when revolving around the y-axis?

Find the y-values where the region begins and ends along the y-axis.

What is the first step in solving a volume of revolution problem?

Determine the axis of revolution (x-axis or y-axis).

How do you handle a problem where the function is not explicitly given?

Derive the function from the given information or geometric properties.

What do you do if you can't directly integrate the squared function?

Use trigonometric substitution, integration by parts, or other integration techniques.

How do you check if your answer is reasonable?

Estimate the volume based on the shape and dimensions of the solid.

How do you set up the integral if you have the region bounded by $y=x^2$ , $x=2$ , and $y=0$ revolved around the x-axis?

The integral is $V = \pi \int_{0}^{2} (x^2)^2 dx$ .

How do you set up the integral if you have the region bounded by $x=y^2$ , $y=1$ , and $x=0$ revolved around the y-axis?

The integral is $V = \pi \int_{0}^{1} (y^2)^2 dy$ .

Formula for volume using the disc method (x-axis)?

$V = \int_{a}^{b} \pi [f(x)]^2 dx$

Formula for volume using the disc method (y-axis)?

$V = \int_{c}^{d} \pi [f(y)]^2 dy$

What is the area of a disc used in the disc method?

$A = \pi r^2$ , where $r = f(x)$ or $f(y)$ .

What is the volume of a single disc when revolving around the x-axis?

$dV = \pi [f(x)]^2 dx$

What is the volume of a single disc when revolving around the y-axis?

$dV = \pi [f(y)]^2 dy$

If $y=x^3$ , express $x$ in terms of $y$ .

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

What is the general form of an integral for volume of revolution?

$V = \pi \int [radius]^2 d(axis)$

How to find the radius of a disc when rotating around the x-axis?

The radius is given by the function $f(x)$ defining the curve.

How to find the radius of a disc when rotating around the y-axis?

The radius is given by the function $f(y)$ defining the curve.

What is the relationship between radius and function when revolving around an axis?

Radius = Function value at a given point on the axis of revolution.

Explain the concept of slicing in the disc method.

Dividing the solid into infinitely thin discs perpendicular to the axis of rotation to approximate its volume.

Why do we square the function $f(x)$ or $f(y)$ in the disc method?

Because we are calculating the area of a circle ( $\pi r^2$ ) where $r$ is the radius, represented by $f(x)$ or $f(y)$ .

What is the relationship between the axis of revolution and the variable of integration?

The variable of integration ( $dx$ or $dy$ ) must be perpendicular to the axis of revolution.

When should you use the disc method?

When the solid of revolution has no holes, and the cross-sections perpendicular to the axis of rotation are discs.

Explain the role of the integral in the disc method.

The integral sums up the volumes of the infinitely thin discs to find the total volume of the solid.

How does the choice of axis affect the function used in the disc method?

Rotating around the x-axis uses $f(x)$ , while rotating around the y-axis requires expressing the function as $f(y)$ .

Why is it important to visualize the solid of revolution?

Visualization helps determine the limits of integration and the correct function to use.

What happens if you choose the wrong axis of integration?

The resulting integral will likely be incorrect, leading to a wrong volume calculation.

Explain how the disc method relates to Riemann sums.

The disc method is a continuous application of Riemann sums, where the discs represent the rectangles in the sum.

How does the disc method simplify volume calculation?

It breaks down a complex 3D shape into simple, manageable discs whose volumes can be easily summed using integration.