Find the exact area of the surface obtained by rotating the curve about the x-axis. x=(1)/(3)(y^

Find the exact area of the surface obtained by rotating the...

Key Concepts

Differentiation using the Chain Rule

In many surface area problems, computing the derivative of the given function accurately is essential. The chain rule is a fundamental tool in differentiation which allows one to differentiate composite functions by taking the derivative of the outer function multiplied by the derivative of the inner function. This process is critical when obtaining the expression under the square root in the surface area integral.

Integration Techniques and Substitution

After setting up the integral for the surface area, the evaluation often requires advanced integration techniques. One common method is the substitution technique, which simplifies the integral by changing variables. This method helps in integrating expressions that are results of chaining functions, particularly when the integrals involve non-linear composite functions.

Surface Area of Solids of Revolution

This concept involves finding the area of a surface created when a curve is revolved around an axis. The formulas differ depending on the axis of rotation and the representation of the curve. When the curve is given in terms of y (or x), the integral is set up by multiplying the circumference element (2? times the distance from the axis) by the differential arc length, which includes the square root of one plus the square of the derivative. Understanding this setup is crucial for solving problems involving surfaces of revolution.

Transcript

00:01 For this problem, we want to find the exact area of the surface obtained by rotating the curve x equals 1 3 3x2 to the power of 3 over 2 about the x axis.

00:11 So to begin, we want to rearrange that equation to get y as a function of x.

00:17 Doing so, we should get that y is equal to 3x to the power of 2 over 3 minus 2.

00:26 Which then means that y prime of x will equal one -half of three x the power of two over three minus two to the power of negative one -half times now we want the derivative of the inside so that would be the derivative of 3 to the power of 2 over 3 x times x to the power of or 3 to the power 2 over 3 times x to the power of 2 over 3 so it'll be 3 to the power of 2 over 3 times 2 over 3 x to the power of 2 over 3 minus 1 so x the power of negative 1 over 3 minus 2 which then means uh one second here the most simplified form that we can get that derivative into would be 1 over 3x to the power 1 over 3 times the square root of 3x or 2 over 3 minus 2 which then means that our y prime squared which will need for our surface area equation is going to come out to 1 over 3x the power of 2 over 3 times 3x to the power of 2 over 3 minus 2 so now that we have that we can write down our surface area equation which will be the integral from 1 to 2 oh actually we do need to be careful because we're trying to get the rotation about the x -axis you need to figure out what x would be when y equals 1 and when y equals 2 so x of 1 should come out to be root 3 and x of 2 should come out to be 2 root 6.

02:39 So we'll have that our surface area is going to equal the integral from root 3 up to 2 root 6 of 2 pi y.

02:50 So 2 pi times the square root of 3x the power of 2 over 3 minus 2 times the square root of 1 plus 1 over 3x to the power of 2 over 3 times the square root of 1 plus 1 over 3 x to the power of 2 over 3 times the square root of 1 plus 1 over 3, times 3x the power of 2 over 3 minus 2 d x.

03:19 So having that, we should now be able to get the indefinite expression for our integral...

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