How do you Use a Riemann sum to find volume?
which can be expressed as the limit of the right Riemann sum
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To use a Riemann sum to find volume, follow these steps:
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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 ).
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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.
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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.
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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.
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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.
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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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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.
- How do you estimate the area under the graph of #f(x) = sqrt x# from #x=0# to #x=4# using four approximating rectangles and right endpoints?
- How do you use Riemann sums to evaluate the area under the curve of #f(x) = 2-x^2# on the closed interval [0,2], with n=4 rectangles using midpoint?
- If the area under the curve of f(x) = 25 – x2 from x = –4 to x = 0 is estimated using four approximating rectangles and left endpoints, will the estimate be an underestimate or overestimate?
- How do you use the trapezoidal rule to approximate the Integral from 0 to 0.5 of #(1-x^2)^0.5 dx# with n=4 intervals?
- How do you find the Riemann sum for this integral using right endpoints and n=3 for the integral #int (2x^2+2x+6)dx# with a = 5 and b = 11?

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