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Calculus II/III

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A. Find the integral of the following function.

Integral of (x√(x+1)) dx.

B. Set up and evaluate the integral of (2√x) for the area of the surface generated by revolving the curve about the x-axis from 4 to 9.

For part B of our question , the surface of revolution is is
integral of 2 pi y * dx
= 4 pi x^(1/2) dx
for x from 4 to 9.
The indefinite integral is
4 pi x^(3/2)/(3/2)
For the definite integral, subtract the value at x = 4 from the value at x = 9.

Try making the subsitution u = x + 1
x = u -1
dx = du
The integral becomes

Integral of (u^3/2) - u^(1/2) du
= (2/5) [u^(5/2) - (2/3) u^(3/2)] du
= (2/5) (x+1)^(5/2) - (2/3)(x+1)^(3/2)]

For the part B of you question, integrate
2 sq

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