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Definite Integrals, Areas, and Basic Anti-Derivatives
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Concept of an integral, areas with definite integrals, and basic anti-derivative solving skills
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Area under a curve
SL 5.5
The area between a curve āf(x)>0ā and theĀ āxā-axis is given by
ā
A=ā«abāf(x)dxš
āIntegration as reverse differentiation
SL 5.5
Integration, or anti-differentiation, is essentially the opposite of differentiation. We use the integral symbolāā«āĀ and write:
ā
ā«f(x)dx=a function with a derivative off(x)
āBy convention we denote this functionĀ āF:
ā
ā«f(x)dx=F(x)š«
āWe can also write
ā
ā«dxdyādx=yš«
āNotice theĀ ādxāĀ under the integral. This tells us which variable we are integrating with respect to - in this case we are reversingĀ ādxdā.
The Integration Constant
SL 5.5
Since the derivative of a constant is always zero, then if ifĀ āFā²(x)=f(x),Ā thenĀ ā(F(x)+C)ā²=f(x).
This means that when we integrate, we can add any constant to our result, since differentiating makes this constant irrelevant:
ā
ā«f(x)dx=F(x)+Cš«
āAnti-Derivative of xāæ, nāā¤
SL 5.5
ā
ā«xndx=n+1xn+1ā+C,nāZ,nī =ā1š
āIntegrals of sums and scalar products
SL 5.5
In the same way that constant multiples can pass through the derivative, they can pass through the integral:
ā
ā«af(x)dx=aā«f(x)dxš«
āAnd in the same way that the derivative of a sum is the sum of the derivatives:
ā
ā«f(x)+g(x)dx=ā«f(x)dx+ā«g(x)dxš«
āBoundary Conditions
SL 5.5
If we know the value ofĀ āyāĀ orĀ āf(x)āĀ for a givenĀ āx,Ā we can determineĀ āCāĀ by plugging inĀ āxāĀ andĀ āy.
Definite Integrals
SL 5.5
A definite integral is evaluated between a lower and upper bound.
ā
ā«abāf(x)dx is the integral of f(x)dx from x=a to x=b
āWe can solve a definite integral with
ā
ā«abāf(x)dx=[F(x)]abā=F(b)āF(a)š«
āwhereĀ āF(x)=ā«f(x)dx.
Calculating Definite Integral with GDC
SL 5.5
Graphing calculators can be used to evaluate definite integrals.
For example, on a TI-84, math > 9:fnInt(, which prompts you withĀ āā«ā”ā”ā(ā”)dā”.Ā Make sure the variable of your function matches the variable that you take the integral with respect to.
Definite Integral Rules
SL 5.5
Integrals of the same function with adjacent domains can be merged:
ā
ā«abāf(x)dx+ā«bcāf(x)dx=ā«acāf(x)dxš«
āSimilarly, the domain of an integral can be split:
ā
ā«abāf(x)dx=ā«amāf(x)dx+ā«mbāf(x)dxš«
āfor anyĀ āa<m<b.
Area between curve and x-axis
SL 5.11
In general, the area enclosed between a curve and theĀ āxā-axis is given by
ā
A=ā«abāā£f(x)ā£dxš
āsince any region below theĀ āxā-axis hasĀ āf(x)<0,Ā but area must always be positive.
This can be done with technology, or by splitting the integral into parts - whereĀ āfāĀ is positive and whereĀ āfāĀ is negative:
ā
A=ā«amāf(x)dx+ā«mbāāf(x)dxš«
āArea between curves
SL 5.11
The area enclosed between two curves is given by
ā
A=ā«abāā£f(x)āg(x)ā£dx
āThis can be done with technology, or by splitting the integral into multiple regions, each having eitherĀ āf(x)>g(x)āĀ orĀ āg(x)>f(x).
Area between curve and y-axis
AHL 5.17
We find the area between a curve and theĀ āyā-axis by taking an integral with respect toĀ āy.Ā Integrals with respect toĀ āyāĀ haveĀ āyā-value upper and lower bounds and functions in terms ofĀ āy.Ā Therefore,
ā
ā«abāā£f(y)ā£dy,
āwhereĀ āaāĀ is the minimumĀ āyā-value andĀ ābāĀ is the maximum value, returns the area between theĀ āyā-axis and a curveĀ āf(y):
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