# What is the area under #f(x)=5x-1# in #x in[0,2] #?

The net area is

Method 1:

The bounded net area is that of a trapezium with heights:

Method 2:

We can use calculus, and evaluate the definite integral:

Note:

Both of the above methods calculate the "net" area, whereas the actual area is somewhat different:

graph{(y-5x+1)(y-10000x)(y-10000x+20000)=0 [-1, 3, -5, 12]}

The actual area is:

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To find the area under the curve of the function f(x) = 5x - 1 in the interval [0, 2], you need to compute the definite integral of the function over that interval.

∫[0,2] (5x - 1) dx

Now, integrate the function with respect to x:

= [5/2 * x^2 - x] evaluated from 0 to 2

Now plug in the upper and lower limits:

= [(5/2 * 2^2 - 2) - (5/2 * 0^2 - 0)]

= [(5/2 * 4 - 2) - (0)]

= [(10 - 2) - (0)]

= (8 - 0)

= 8

So, the area under the curve of f(x) = 5x - 1 in the interval [0, 2] is 8 square units.

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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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