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Definite Integrals, Areas, and Basic Anti-Derivatives
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Definite integrals and their rules, using anti-derivatives and the constant of integration, evaluating areas under a curve, between a curve and the x-axis, and between two curves, with ∫abf(x)dx=[F(x)]ab,
Area under a curve
SL 5.5
The area between a curve f(x)>0 and the x-axis is given by
A=∫abf(x)dx📖
Trapezoidal Rule
SL AI 5.8
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 right side of the trapezoid:
The formula for the area using the trapezoid rule approximation is
∫abydx≈21h((y0+yn)+2(y1+y2+⋯+yn−1))
where h=nb−a.
Example: Approximate ∫03exdx using n=6 trapezoids.
Recognize that b=3, a=0 and n=6, so the width of each trapezoid is
h=63−0=0.5
Using the formula, the area is
∫03exdx≈21⋅0.5⋅(y0+y6+2(y1+⋯+y5))
Each yk=e0.5×k, but thankfully we can use our calculator here:
In L1 enter 0,1…6.
At the top of L2, set L2=e0.5L1 and hit enter. The values fill in:
Now (y0+y6+2(y1+⋯+y5)) is
1+20.086+2(1.6487+2.7183+⋯+12.182)≈78.6252
So the area is
∫03exdx≈21⋅0.5⋅78.6252≈19.7
Integration as reverse differentiation
SL 5.5
Integration, or anti-differentiation, is essentially the opposite of differentiation. We use the integral symbol∫ and write:
∫f(x)dx=a function with a derivative off(x)
By convention we denote this function F:
∫f(x)dx=F(x)🚫
We can also write
∫dxdydx=y🚫
Notice the dx under the integral. This tells us which variable we are integrating with respect to - in this case we are reversing dxd.
The Integration Constant
SL 5.5
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:
∫f(x)dx=F(x)+C🚫
Anti-Derivative of xⁿ, n∈ℤ
SL 5.5
∫xndx=n+1xn+1+C,n∈Z,n=−1📖
Integrals 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:
∫af(x)dx=a∫f(x)dx🚫
And in the same way that the derivative of a sum is the sum of the derivatives:
∫f(x)+g(x)dx=∫f(x)dx+∫g(x)dx🚫
Boundary Conditions
SL 5.5
If we know the value of y or f(x) for a given x, we can determine C by plugging in x and y.
Definite Integrals
SL 5.5
A definite integral is evaluated between a lower and upper bound.
∫abf(x)dx is the integral of f(x)dx from x=a to x=b
We can solve a definite integral with
∫abf(x)dx=[F(x)]ab=F(b)−F(a)🚫
where F(x)=∫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 with ∫□□(□)d□. 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:
∫abf(x)dx+∫bcf(x)dx=∫acf(x)dx🚫
Similarly, the domain of an integral can be split:
∫abf(x)dx=∫amf(x)dx+∫mbf(x)dx🚫
for any a<m<b.
Area between curve and x-axis
AHL AI 5.11
In general, the area enclosed between a curve and the x-axis is given by
A=∫ab∣f(x)∣dx📖
since any region below the x-axis has f(x)<0, but area must always be positive.
This can be done with technology, or by splitting the integral into parts - where f is positive and where f is negative:
A=∫amf(x)dx+∫mb−f(x)dx🚫
Area between curves
AHL AI 5.11
The area enclosed between two curves is given by
A=∫ab∣f(x)−g(x)∣dx
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).
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