# What is the volume of the solid produced by revolving #f(x)=x^2+3x-sqrtx, x in [0,3] #around the x-axis?

601.5 cubic units, nearly

The two x-scaled and y-scaled graphs reveal that the said

shuttlecock-like solid of revolution has two parts, having just the x-

near the origin might appear as a knot. This one has a convex

surface, in contrast to the other that has concave surface.

The volume is

for the limits

between x =. 0 and 3

#=601.5 cubic units, nearly.

graph{0.1(x^2+3x-sqrtx) [-10, 10, -5, 5]} graph{(10(x^2+3x-sqrtx)) [-0.1, 0.1, -10, 10]}

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To find the volume of the solid produced by revolving ( f(x) = x^2 + 3x - \sqrt{x} ) around the x-axis over the interval ([0,3]), you use the disk method or washer method.

The volume ( V ) is given by the integral:

[ V = \pi \int_{0}^{3} [f(x)]^2 , dx ]

Compute ( f(x) = x^2 + 3x - \sqrt{x} ) and square it. Then integrate the squared function over the interval ([0,3]) using definite integration.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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