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Definite Integrals, Areas, and Basic Anti-Derivatives
Concept of an integral, areas with definite integrals, and basic anti-derivative solving skills
Concept of an integral, areas with definite integrals, and basic anti-derivative solving skills
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The area between a curve f(x)>0 and the x-axis is given by
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Approximate area beneath a curve from x=a to x=b by summing the area of n trapezoids of equivalent width.
Each trapezoid's area is the common width (nb−a) times the average of the function's value on the left and rigth side of the trapezoid:
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Integration, or anti-differentiation, is essentially the opposite of differentiation. We use the integral symbol∫ and write:
By convention we denote this function F:
We can also write
Notice the dx under the integral. This tells us which variable we are integrating with respect to - in this case we are reversing dxd.
Since the derivative of a constant is always zero, then if if F′(x)=f(x), then (F(x)+C)′=f(x).
This means that when we integrate, we can add any constant to our result, since differentiating makes this constant irrelevant:
In the same way that constant multiples can pass through the derivative, they can pass through the integral:
And in the same way that the derivative of a sum is the sum of the derivatives:
If we know the value of y or f(x) for a given x, we can determine C by plugging in x and y.
A definite integral is evaluated between a lower and upper bound.
We can solve a definite integral with
where F(x)=∫f(x)dx.
Graphing calculators can be used to evaluate definite integrals.
For example, on a TI-84, math > 9:fnInt(, which prompts you with ∫□□(□)d□. Make sure the variable of your function matches the variable that you take the integral with respect to.
Integrals of the same function with adjacent domains can be merged:
Similarly, the domain of an integral can be split:
for any a<m<b.
In general, the area enclosed between a curve and the x-axis is given by
since any region below the x-axis has f(x)<0, but area must always be positive.
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This can be done with technology, or by splitting the integral into parts - where f is positive and where f is negative:
The area enclosed between two curves is given by
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This can be done with technology, or by splitting the integral into multiple regions, each having either f(x)>g(x) or g(x)>f(x).