How do you Use a Riemann sum to find volume?

Answer 1
If you have a cross-sectional area function #A(x)# of the solid that spans from #x=a# to #x=b#, then you can find the volume #V# by
#V=int_a^b A(x)dx#,

which can be expressed as the limit of the right Riemann sum

#=lim_{n to infty}sum_{i=1}^infty A(a+iDeltax)Delta x#,
where #Delta x={b-a}/n#
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Answer 2

To use a Riemann sum to find volume, follow these steps:

  1. Determine the region: Identify the region in the xy-plane bounded by a function ( f(x) ), the x-axis, and two vertical lines ( x = a ) and ( x = b ).

  2. Choose partitions: Divide the interval ( [a, b] ) into ( n ) subintervals of equal width ( \Delta x = \frac{b - a}{n} ). Each subinterval is ( \Delta x ) wide.

  3. Select sample points: Choose sample points ( x_i^* ) in each subinterval ( [x_{i-1}, x_i] ), where ( x_{i-1} ) and ( x_i ) are the endpoints of the ( i )th subinterval.

  4. Form Riemann sum: The Riemann sum is given by ( \sum_{i=1}^{n} f(x_i^*) \cdot \Delta A_i ), where ( \Delta A_i ) is the area of a representative rectangle or slice in the region.

  5. Volume approximation: The volume of the solid formed by revolving the region about a horizontal or vertical axis is approximately given by the Riemann sum multiplied by the thickness of the slices or the height of the representative rectangles.

  6. Take the limit: As ( n ) approaches infinity, the Riemann sum approaches the definite integral ( \int_{a}^{b} f(x) , dx ). Therefore, the volume can be found by evaluating the definite integral.

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Answer from HIX Tutor

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.

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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