How do you approximate the area under #y=10−x^2# on the interval [1, 3] using 4 subintervals and midpoints?
First you divide the interval in four equal parts, and in each part you find the midpoint.
You do the same for the other three parts and add. (Of course you round the total, as 5 decimals is a bit steep for an approxiation!)
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To approximate the area under ( y = 10  x^2 ) on the interval [1, 3] using 4 subintervals and midpoints, you can follow these steps:

Determine the width of each subinterval: [ \Delta x = \frac{b  a}{n} = \frac{3  1}{4} = \frac{2}{4} = 0.5 ]

Find the midpoints of each subinterval:
 Subinterval 1: Midpoint = 1 + 0.5/2 = 1.25
 Subinterval 2: Midpoint = 1.75
 Subinterval 3: Midpoint = 2.25
 Subinterval 4: Midpoint = 2.75

Evaluate the function ( y = 10  x^2 ) at each midpoint to get the corresponding yvalues:
 At x = 1.25: ( y = 10  (1.25)^2 = 8.4375 )
 At x = 1.75: ( y = 10  (1.75)^2 = 6.4375 )
 At x = 2.25: ( y = 10  (2.25)^2 = 3.4375 )
 At x = 2.75: ( y = 10  (2.75)^2 = 0.4375 )

Calculate the area of each rectangle formed by the midpoints and the corresponding yvalues:
 Area 1 = ( 0.5 \times 8.4375 = 4.21875 )
 Area 2 = ( 0.5 \times 6.4375 = 3.21875 )
 Area 3 = ( 0.5 \times 3.4375 = 1.71875 )
 Area 4 = ( 0.5 \times 0.4375 = 0.21875 )

Add up the areas of all four rectangles to approximate the total area under the curve: [ \text{Total Area} = \text{Area 1} + \text{Area 2} + \text{Area 3} + \text{Area 4} = 4.21875 + 3.21875 + 1.71875 + 0.21875 = 9.375 ]
Therefore, the approximate area under the curve ( y = 10  x^2 ) on the interval [1, 3] using 4 subintervals and midpoints is 9.375 square units.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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