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Calculus AB : Finding Volume Using Integration

Study concepts, example questions & explanations for Calculus AB

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Example Questions

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Example Question #1 : Apply The Revolving Disc Method And Washer Method

What is the disc method?

Possible Answers:

Finding the volume of a sphere

Finding the volume of a solid by taking a cross sectional of it and calculating the volume by adding infinitely many of these cross sections

Finding the volume of a solid by finding the surface area and adding all surface areas of the solid

Finding the area of a circle

Correct answer:

Finding the volume of a solid by taking a cross sectional of it and calculating the volume by adding infinitely many of these cross sections

Explanation:

The disc method allows us to find the volume of a solid in a 2 dimensional space by taking the cross section of a solid. When taking the cross section, you are slicing perpendicular to the axis of rotation (usually either x or y axis). Then we sum infinitely many of these cross sections to obtain a value for the volume, sort of like integrating under the curve.

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Example Question #2 : Apply The Revolving Disc Method And Washer Method

What is the formula for the disc method?

Possible Answers:

$V = \pi\int _a^b f(x)^2 dx$

$V = \pi r^2 h$

$V = \pi r^2\int_a^b f(x)^2 dx$

$V = \int_a^b \pi r^2 h dr$

Correct answer:

$V = \pi\int _a^b f(x)^2 dx$

Explanation:

The disc method is based on the volume of a cylinder, $V = \pi r^2 h$ . In the formula for the disc method, is the volume of the solid, is the smallest value of of f(x) , is the largest value of of f(x) , f(x) is the radius of the disc, and is the height of the disc.

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Example Question #3 : Apply The Revolving Disc Method And Washer Method

Find the volume of the solid that is bounded by y = x^2 - 2x + 1 and the x-axis about the x-axis when $0\leq x\leq 2$ .

Possible Answers:

$V = \frac{2\pi}{9}$

$V = \frac{\pi}{9}$

$V = -2\pi$

$V = 2\pi$

Correct answer:

$V = \frac{2\pi}{9}$

Explanation:

First we must find the cross sectional area. To do this, we need the distance from the x-axis to the function (the radius of each disc). If we think about it, this is just the function itself.

$A(x) = \pi f(x)^2$

$A(x) = \pi(x^2-2x+1)^2$

Now we can plug into our disc method formula: $V = \int_a^b A(x) dx$ or $V = \pi\int_a^b f(x)^2dx$ .

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

$V = \pi\int_0^2 (x^2 - 2x + 1)^2 dx$

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

$V = \pi\int_0^2 (x-1)^8 dx$

Use a u-substitution where u = (x-1), du = 1

$V = \pi\int_0^2 u^8 du$

$V = \pi[\frac{u^9}{9}]_0^2$

$V = \pi[\frac{(x-1)^9}{9}]_0^2$

$V = \pi[\frac{(2-1)^9}{9}-\frac{(0-1)^9}{9}]$

$V = \pi[\frac{1}{9} + \frac{1}{9}]$

$V = \frac{2\pi}{9}$

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Example Question #4 : Apply The Revolving Disc Method And Washer Method

Find the volume of the solid using the disc method bounded by y = 5x^3 - 4x and the x-axis when $-1\leq x \leq 4$ about the x-axis. Do not simplify.

Possible Answers:

$V = \pi$

$V = \pi\frac{409625}{7}$

$V = \pi80921$

$V = \pi[\frac{409625}{7} - 7897]$

Correct answer:

$V = \pi[\frac{409625}{7} - 7897]$

Explanation:

First set up the disc method formula then plug in the given function.

$V = \pi\int_a^b f(x)^2 dx$

$V = \pi\int_{-1}^{4} (5x^3 - 4x)^2 dx$

$V = \pi\int_{-1}^{4} 25x^6-40x^4 +14x^2 dx$

$V = \pi[\frac{25x^7}{7} -8x^5 + \frac{14x^3}{3}]_{-1}^{4}$

$V = \pi[(\frac{25(4)^7}{7} - 8(4)^5 + \frac{14(4)^3}{3}) -(\frac{25(-1)^7}{7} - 8(-1)^5 + \frac{14(-1)^3}{3})$

$V = \pi[\frac{409600}{7} - 8193 + \frac{896}{3} +\frac{25}{7} -8 + \frac{14}{3}]$

$V = \pi[\frac{409625}{7} - 8201 + \frac{912}{3}]$

$V = \pi[\frac{409625}{7} - 7897]$

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Example Question #2 : Apply The Revolving Disc Method And Washer Method

True or False: The Washer Method is more accurate than the Disc Method and should be used all the time.

Possible Answers:

False

True

Correct answer:

False

Explanation:

These two methods are used in different situations so their accuracy is not comparable. The Washer Method is like using the disc method when considering the difference between two discs. Their accuracy is similar and you should use the Washer method when you have an inner and outer radius and the disc method when you have only one radius.

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Example Question #1 : Apply The Revolving Disc Method And Washer Method

What is the disc method?

Possible Answers:

Finding the volume of a sphere

Finding the volume of a solid by taking a cross sectional of it and calculating the volume by adding infinitely many of these cross sections

Finding the volume of a solid by finding the surface area and adding all surface areas of the solid

Finding the area of a circle

Correct answer:

Finding the volume of a solid by taking a cross sectional of it and calculating the volume by adding infinitely many of these cross sections

Explanation:

Report an Error

Example Question #2 : Apply The Revolving Disc Method And Washer Method

What is the formula for the disc method?

Possible Answers:

$V = \pi\int _a^b f(x)^2 dx$

$V = \pi r^2 h$

$V = \pi r^2\int_a^b f(x)^2 dx$

$V = \int_a^b \pi r^2 h dr$

Correct answer:

$V = \pi\int _a^b f(x)^2 dx$

Explanation:

Report an Error

Example Question #3 : Apply The Revolving Disc Method And Washer Method

Find the volume of the solid that is bounded by y = x^2 - 2x + 1 and the x-axis about the x-axis when $0\leq x\leq 2$ .

Possible Answers:

$V = \frac{2\pi}{9}$

$V = \frac{\pi}{9}$

$V = -2\pi$

$V = 2\pi$

Correct answer:

$V = \frac{2\pi}{9}$

Explanation:

First we must find the cross sectional area. To do this, we need the distance from the x-axis to the function (the radius of each disc). If we think about it, this is just the function itself.

$A(x) = \pi f(x)^2$

$A(x) = \pi(x^2-2x+1)^2$

Now we can plug into our disc method formula: $V = \int_a^b A(x) dx$ or $V = \pi\int_a^b f(x)^2dx$ .

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

$V = \pi\int_0^2 (x^2 - 2x + 1)^2 dx$

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

$V = \pi\int_0^2 (x-1)^8 dx$

Use a u-substitution where u = (x-1), du = 1

$V = \pi\int_0^2 u^8 du$

$V = \pi[\frac{u^9}{9}]_0^2$

$V = \pi[\frac{(x-1)^9}{9}]_0^2$

$V = \pi[\frac{(2-1)^9}{9}-\frac{(0-1)^9}{9}]$

$V = \pi[\frac{1}{9} + \frac{1}{9}]$

$V = \frac{2\pi}{9}$

Report an Error

Example Question #4 : Apply The Revolving Disc Method And Washer Method

Find the volume of the solid using the disc method bounded by y = 5x^3 - 4x and the x-axis when $-1\leq x \leq 4$ about the x-axis. Do not simplify.

Possible Answers:

$V = \pi$

$V = \pi\frac{409625}{7}$

$V = \pi80921$

$V = \pi[\frac{409625}{7} - 7897]$

Correct answer:

$V = \pi[\frac{409625}{7} - 7897]$

Explanation:

First set up the disc method formula then plug in the given function.

$V = \pi\int_a^b f(x)^2 dx$

$V = \pi\int_{-1}^{4} (5x^3 - 4x)^2 dx$

$V = \pi\int_{-1}^{4} 25x^6-40x^4 +14x^2 dx$

$V = \pi[\frac{25x^7}{7} -8x^5 + \frac{14x^3}{3}]_{-1}^{4}$

$V = \pi[(\frac{25(4)^7}{7} - 8(4)^5 + \frac{14(4)^3}{3}) -(\frac{25(-1)^7}{7} - 8(-1)^5 + \frac{14(-1)^3}{3})$

$V = \pi[\frac{409600}{7} - 8193 + \frac{896}{3} +\frac{25}{7} -8 + \frac{14}{3}]$

$V = \pi[\frac{409625}{7} - 8201 + \frac{912}{3}]$

$V = \pi[\frac{409625}{7} - 7897]$

Report an Error

Example Question #2 : Apply The Revolving Disc Method And Washer Method

True or False: The Washer Method is more accurate than the Disc Method and should be used all the time.

Possible Answers:

False

True

Correct answer:

False

Explanation:

Report an Error

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Calculus AB : Finding Volume Using Integration

Study concepts, example questions & explanations for Calculus AB

All Calculus AB Resources

Example Questions

Example Question #1 : Apply The Revolving Disc Method And Washer Method

Example Question #2 : Apply The Revolving Disc Method And Washer Method

Example Question #3 : Apply The Revolving Disc Method And Washer Method

Example Question #4 : Apply The Revolving Disc Method And Washer Method

Example Question #2 : Apply The Revolving Disc Method And Washer Method

Example Question #1 : Apply The Revolving Disc Method And Washer Method

Example Question #2 : Apply The Revolving Disc Method And Washer Method

Example Question #3 : Apply The Revolving Disc Method And Washer Method

Example Question #4 : Apply The Revolving Disc Method And Washer Method

Example Question #2 : Apply The Revolving Disc Method And Washer Method

All Calculus AB Resources

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