Question 1

Find the area of the surface generated when y = x³ from x = 0 to x = 1 is rotated around the x-axis.

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Question 2

Find the arc length of the curve y = 2x^3/2 from x = 0 to x = 5. Approximate area to two decimal places.

Accepted Answer

The given curve is $y = 2x^{3/2}$. To find the arc length from $x = 0$ to $x = 5$, we use the formula for arc length in terms of $x$, which is $L = \int_{a}^{b} \sqrt{1 + (\frac{dy}{dx})^2} dx$. First, we find the derivative of $y$ with respect to $x$, which is $\frac{dy}{dx} = 3x^{1/2}$. Plugging this into the formula gives us an integral that, when evaluated from $x = 0$ to $x = 5$, yields an approximate value of 23.04. This calculation involves integrating the square root of $1 + (3x^{1/2})^2$, simplifying, and then evaluating the definite integral.

Question 3

Find the arc length of the curve  from x = 0 to x = 3.

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Question 4

Answer true or false. The arc length of the curve  from x = 5 to x = 7 is 2.084.

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As the given curve is not visible, we cannot verify the solution but we assume it to be true.

Question 5

The arc length of the curve  from y = 0 to y = 1 is

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Question 6

Find the area of the surface generated when  from x = 3 to x = 6 is revolved about the x-axis.

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Question 7

Find the area of the surface generated when  from x = 1 to x = 3 is rotated around the x-axis.

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Question 8

Find the area of the surface generated when  from y = 0 to y = 3 is rotated around the y-axis. Approximate area to three decimal places.

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Question 9

Find the area of the surface generated when y = 8 - x from x = 0 to x = 6 is rotated around the x-axis.

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Question 10

Find the area of the surface that is generated by revolving the portion of the curve y = x⁴ between x = 0 and x = 2 about the x-axis.

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Question 11

Find the area of the surface generated when  from x = 0 to x = 4 is rotated around the x-axis.

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Question 12

Find the area of the surface generated when  from x = 3 to x = 6 is rotated around the x-axis.

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Question 13

Find the area of the surface of a sphere of radius r which is generated by rotating a semicircle about a diameter.

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Question 14

Find the area of the surface generated when y² = 12x from x = 0 to x = 1 is rotated around the x-axis. Approximate area to three decimal places.

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Question 15

Answer true or false. The arc length of the curve y = x³ from x = 0 to x = 3 is 27.658.

Accepted Answer

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Question 16

Find the area of the surface generated when  from y = 1 to y = 2 is rotated around the y-axis.

Accepted Answer

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Question 17

Find the area of the surface that is generated by revolving the portion of the curve y = x³ between x = 0 and x = 2 about the y-axis. Approximate area to three decimal places.

Accepted Answer

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Question 18

Answer true or false. The arc length of the curve y = (x - 1)^7/2 from x = 0 to x = 2 is given by .

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Question 19

Find the area of the surface that is generated by revolving the portion of the curve y = x² - 2 between x = 0 and x = 1 about the x-axis. Approximate area to three decimal places.

Accepted Answer

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Question 20

Find the area of the surface generated when  from x = 1 to x = 3 is rotated around the x-axis.

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Quiz 6: Applications of the Definite Integral in Geometry, Science and Engineering

Find the area of the surface generated when y = x3 from x = 0 to x = 1 is rotated around the x-axis.

Find the arc length of the curve y = 2x3/2 from x = 0 to x = 5. Approximate area to two decimal places.

Find the arc length of the curve from x = 0 to x = 3.

Answer true or false. The arc length of the curve from x = 5 to x = 7 is 2.084.

The arc length of the curve from y = 0 to y = 1 is

Find the area of the surface generated when from x = 3 to x = 6 is revolved about the x-axis.

Find the area of the surface generated when from x = 1 to x = 3 is rotated around the x-axis.

Find the area of the surface generated when from y = 0 to y = 3 is rotated around the y-axis. Approximate area to three decimal places.

Find the area of the surface generated when y = 8 - x from x = 0 to x = 6 is rotated around the x-axis.

Find the area of the surface that is generated by revolving the portion of the curve y = x4 between x = 0 and x = 2 about the x-axis.

Find the area of the surface generated when from x = 0 to x = 4 is rotated around the x-axis.

Find the area of the surface generated when from x = 3 to x = 6 is rotated around the x-axis.

Find the area of the surface of a sphere of radius r which is generated by rotating a semicircle about a diameter.

Find the area of the surface generated when y2 = 12x from x = 0 to x = 1 is rotated around the x-axis. Approximate area to three decimal places.

Answer true or false. The arc length of the curve y = x3 from x = 0 to x = 3 is 27.658.

Find the area of the surface generated when from y = 1 to y = 2 is rotated around the y-axis.

Find the area of the surface that is generated by revolving the portion of the curve y = x3 between x = 0 and x = 2 about the y-axis. Approximate area to three decimal places.

Answer true or false. The arc length of the curve y = (x - 1)7/2 from x = 0 to x = 2 is given by .

Find the area of the surface that is generated by revolving the portion of the curve y = x2 - 2 between x = 0 and x = 1 about the x-axis. Approximate area to three decimal places.

Find the area of the surface generated when from x = 1 to x = 3 is rotated around the x-axis.

Find the area of the surface generated when y = x³ from x = 0 to x = 1 is rotated around the x-axis.

Find the arc length of the curve y = 2x^3/2 from x = 0 to x = 5. Approximate area to two decimal places.

Find the area of the surface that is generated by revolving the portion of the curve y = x⁴ between x = 0 and x = 2 about the x-axis.

Find the area of the surface generated when y² = 12x from x = 0 to x = 1 is rotated around the x-axis. Approximate area to three decimal places.

Answer true or false. The arc length of the curve y = x³ from x = 0 to x = 3 is 27.658.

Find the area of the surface that is generated by revolving the portion of the curve y = x³ between x = 0 and x = 2 about the y-axis. Approximate area to three decimal places.

Answer true or false. The arc length of the curve y = (x - 1)^7/2 from x = 0 to x = 2 is given by .

Find the area of the surface that is generated by revolving the portion of the curve y = x² - 2 between x = 0 and x = 1 about the x-axis. Approximate area to three decimal places.