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write and evaluate the definite integral that represents the volume of the solid formed by revolving the region about the y-axis. y = 25 − x2

User IndigoFire
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Final answer:

To find the volume of the solid formed by revolving the region about the y-axis, we use the method of cylindrical shells. The formula to find the volume is V = 2π * ∫(x * f(x)) dx. We can express the function y = 25 - x^2 in terms of x, substitute it into the formula, and then evaluate the definite integral.

Step-by-step explanation:

To find the volume of the solid formed by revolving the region about the y-axis, we can use the method of cylindrical shells. The formula for finding the volume using cylindrical shells is given by:

V = 2π ∫ (x * f(x)) dx

In this case, the function is y = 25 - x^2. To express the function in terms of x, we solve for x in terms of y: x = ±sqrt(25 - y). Substituting this into the formula, we have:

V = 2π ∫ (±sqrt(25 - y)) * y dy

Now we can evaluate the definite integral by finding the antiderivative of the integrand and then using the limits of integration.

User Jason Galuten
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