Integration

Basic Integration Rules

Let k be a constant

(∫_^)(0*d(x))=C

(∫_^)(k*d(x))=k*x+C

(∫_^)(xn*d(x))=(x(n+1))/(n+1)+C,n≠-1

(∫_^)(cos(x)*d(x))=sin(x)+C

(∫_^)(sin(x)*d(x))=-cos(x)+C

(∫_^)((sec^2)(x)*d(x))=tan(x)+C

(∫_^)(sec(x)*tan(x)*d(x))=sec(x)+C

(∫_^)((csc^2)(x)*d(x))=-cot(x)+C

(∫_^)(csc(x)*cot(x)*d(x))=-csc(x)+C

Definite Integrals (The Fundamental Theorem of Calculus)

A definite integral is an integral with upper and lower limits, a and b, respectively, that define a specific interval on the graph. A definite integral is used to find the area bounded by the curve and an axis on the specified interval (a,b)

If F(x) is the antiderivative of a continuous function ƒ(x),the evaluation of the definite integral to calculate the area on the specified interval (a,b) is the First Fundamental Theorem of Calculus:

(∫_a^b)(ƒ(x)*d(x))=F(b)-F(a)

Integration Rules for Definite Integrals

(∫_a^a)(ƒ(x))=0

(∫_a^b)(ƒ(x)*d(x))=(∫_a^c)(ƒ(x)*d(x))+(∫_c^b)(ƒ(x)*d(x)) for c is a value of x, lying between a and b

(∫_a^b)(ƒ(x)*d(x))=-(∫_b^a)(ƒ(x)*d(x))

(∫_-a^a)(ƒ(x)*d(x))=2*(∫_0^a)(ƒ(x)*d(x)), where ƒ is an even function

(∫_-a^a)(ƒ(x)*d(x))=0, where f is an odd function

Riemann Sum (Approximations)

A Riemann Sum is the use of geometric shapes (rectangles and trapezoids) to approximate the area under a curve, therefore approximating the value of a definite integral.

If the interval [a,b] is partitioned into n subintervals, then each subinterval, Δ(x), has a width: Δ(x)=(b-a)/n

Therefore, you find the sum of the geometric shapes, which approximates the area by the following formulas:

Right Riemann Sum:

Area ≌Δ(x)*[ƒ((x_0))+ƒ((x_1))+ƒ((x_2))+⋯+ƒ((x_n-1))]

Left Riemann Sum:

Area ≌ Δ(x)*[ƒ((x_1))+ƒ((x_2))+ƒ((x_3))+⋯+ƒ((x_n))]

Midpoint Riemann Sum

Area ≌ Δ(x)*[ƒ((x_1/2))+ƒ((x_3/2))+ƒ((x_5/2))+⋯+ƒ((x_(2*n-1)/2))]

Trapezoidal Sum

Area ≌ 1/2*Δ(x)*[ƒ((x_0))+2*ƒ((x_1))+2*ƒ((x_2))+⋯2*ƒ((x_n-1))+ƒ((x_n))]

Properties of Riemann Sums

A. The area under the curve is under approximated when

I. A Left Riemann sum is used on an increasing function.

II. A Right Riemann sum is used on a decreasing function.

III. A Trapezoidal sum is used on a concave down function.

B. The area under the curve is over approximated when

I. A Left Riemann sum is used on a decreasing function.

II. A Right Riemann sum is used on an increasing function.

III. A Trapezoidal sum is used on a concave up function.

Riemann Sum (Limit Definition of Area)

Let ƒ be a continuous function on the interval [a,b]. The area of the region bounded by the graph of the function ƒ and the (x^′*′) – axis (i.e. the value of the definite integral) can be found using

(∫_a^b)(ƒ(x)*d(x))=(lim_n→∞)((∑_i=1^n)(ƒ((c_i))*Δ(x)))

where (c_i) is either the left endpoint ((c_i))=a+(i-1)*Δ(x)) or right endpoint ((c_i)=a+i*Δ(x)) and Δ(x) =(b-a)/n

Average Value of a Function

If a function ƒ is continuous on the interval [a,b], the average value of that function ƒ is given by

1/(b-a)*(∫_a^b)(ƒ(x)*d(x))

Second Fundamental Theorem of Calculus

If a function ƒ is continuous on the interval [a,b], let u represent a function of x, then

A. d([(∫_a^x)(ƒ(t)*d(t))])/d(x)=ƒ(x)

B. d([(∫_x^b)(ƒ(t)*d(t))])/d(x)=-ƒ(x)

C. d([(∫_a^u(x))(ƒ(t)*d(t))])/d(x)=ƒ(u(x))⋅(u^′)((x^))

Second Fundamental Theorem of Calculus

If a function f is continuous on the interval [a,b], let u represent a function of x, then

A. d([(∫_a^x)(ƒ(t)*d(t))])/d(x)=ƒ(t)

B. d([(∫_x^b)(ƒ(t)*d(t))])/d(x)=-ƒ(x)

C. d([(∫_a^u(x))(ƒ(t)*d(t))])/d(x)=ƒ(u(x))⋅(u^′)((x^))

Integration of Exponential and Logarithmic Formulas

(∫_^)(1/x*d(x))=ln(|x|)+C

(∫_^)((u^′)/u*d(u))=ln(|u|)+C, where u is a differentiable function of x

(∫_^)(1/(x-a)*d(x))=ln(|x-a|)+C, where a is a constant

(∫_^)(ax*d(x))=(ax)/ln(a)+C

(∫_^)(ex*d(x))=(ex)/ln(e)+C=ex+C

(∫_^)(a)u(x)*d(x)=(au(x))/((ln(a))*(u^′))+C

(∫_^)(eu(x)*d(x))=(eu(x))/(u^′)+C

Integration of Trig and Inverse Trig

(∫_^)(cos(x)*d(x))=sin(x)+C

(∫_^)(sin(x)*d(x))=-cos(x)+C

(∫_^)((sec^2)(x)*d(x))=tan(x)+C

(∫_^)((csc^2)(x)*d(x))=-cot(x)+C

(∫_^)(sec(x)*tan(x)*d(x))=sec(x)+C

(∫_^)(csc(x)*cot(x)*d(x))=-csc(x)+C

(∫_^)(1/√(,1-x2)*d(x))=sec(x)+C

(∫_^)(-1/√(,1-x2)*d(x))=arccosh(x)+C

(∫_^)(1/(1+x2)*d(x))=arctanh(x)+C

(∫_^)(-1/(1+x2)*d(x))=arccoth(x)+C

(∫_^)(1/(|x|√(,x2-1))*d(x))=arcsech(x)+C

(∫_^)(-1/(|x|√(,x2-1))*d(x))=arccsch(x)+C

If u is a differentiable function of x, then

(∫_^)((u^′)/u*d(u))=ln(|u|)+C

(∫_^)((u^′)/√(,1-u2))*d(u)=arcsinh(u)+C

(∫_^)((u^′)/(1+u2)*d(u))=arctanh(u)+C

BC Only: Integration by Parts

If u and v are differentiable functions of x, then

(∫_^)(u*d(v))=u*v-(∫_^)(v*d(u))

BC Only: Partial Fractions

Let R(x) represent a rational function of the form R(x)=N(x)/D(x). If D(x) is a factorable polynomial, Partial Fractions can be used to rewrite R(x) as the sum or difference of simpler rational functions. Then, integration using natural log.

BC Only: Improper Integrals

An improper integral is characterized by having a limits of integration that is infinite or the function ƒ having an infinite discontinuity (asymptote) on the interval [a,b]

Infinite Upper Limit (continuous function)

(∫_a^∞)(ƒ(x)*d(x))=(lim_b→∞)((∫_a^b)(ƒ(x)*d(x)))

Infinite Lower Limit (continuous function)

(∫_-∞^b)(ƒ(x)*d(x))=(lim_a→-∞)((∫_a^b)(ƒ(x)*d(x)))

Both Infinite Limits (continuous function)

(∫_-∞^∞)(ƒ(x)*d(x))=(lim_a→-∞)((∫_a^c)(ƒ(x)*d(x)))+(lim_b→∞)((∫_c^b)(ƒ(x)*d(x))), where c is an x value anywhere on ƒ

Infinite Discontinuity (Let x=k represent an infinite discontinuity on [a,b])

(∫_a^b)(ƒ(x)*d(x))=(lim_x→k-)((∫_a^k)(ƒ(x)*d(x)))+(lim_x→k+)((∫_k^b)(ƒ(x)*d(x)))

BC Only: Arc Length (Length of a Curve)

If the function y=ƒ(x) is a differentiable function, then the length of the arc on [a,b] is

(∫_a^b)(√(,1+[(ƒ^′)(x)]2)*d(x))

If the function x=ƒ(y) a differentiable function, then the length of the arc on [a,b] is

(∫_a^b)(√(,1+[(ƒ^′)(y)]2)*d(y))

Parametric Arc Length: If a smooth curve is given by x(t) and y(t), then the arc length over the interval a≤t≤b is

(∫_a^b)(√(,(d(x)/d(t))2+(d(y)/d(t))2)*d(t))

Exponential Growth and Decay

When the rate of change of a variable y is directly proportional to the value of y, the function y=ƒ(x) is said to grow or decay exponentially

Differential Equation for rate of change: d(y)/d(t)=k*y

General Solution: y=C*e(k*t)

If k>0, then exponential growth occurs

If k<0, then exponential decay occurs

BC Only: Logistic Growth

A population P that experiences a limit factor in the growth of the population based upon the available resources to support the population is said to experience logistic growth

Differential Equation: d(P)/d(t)=k*P*(1-P/L)

General Solution: P(t)=L/(1+b*e(-k*t))

P=population

k=constant growth factor

L=carrying capacity

t=time

b=constant (found with initial condition)

Characteristics of Logistics

The population is growing the fastest where P=L/2

The point where P=L/2 represents a point of inflection

(lim_t→∞)(P(t))=L

Area Between Two Curves

Let y=ƒ(x) and y=g(x) represent two functions such that ƒ(x)≥g(x) (meaning the function ƒ is always above g on the graph) for every x on the interval [a,b]

Area Between Curves =(∫_a^b)([ƒ(x)-g(x)]*d(x))

Let x=ƒ(y) and x=g(y) represents two functions such that ƒ(y)≧g(y) (meaning the function ƒ is always to the right of the function g on the graph) for every y on the interval [a,b]

Area Between Curves = (∫_a^b)([ƒ(y)-g(y)]*d(y))

Volumes of a Solid of Revolution: Disk Method

If a defined region, bounded by a differentiable function ƒ, on a graph is rotated about a line, the resulting solid is called a solid of revolution and the line is called the axis of revolution. The disk method is used when the defined region boarders the axis of revolution over the entire interval [a,b]

Revolving around the x - axis: Volume =π*(∫_a^b)((ƒ(x))2*d(x))

Revolving around the y - axis: Volume =π*(∫_a^b)((ƒ(y))2*d(y))

Revolving around a horizontal line y=k: Volume =π*(∫_a^b)((ƒ(x)-k)2*d(x))

Revolving around a vertical line x=m: Volume =π*(∫_a^b)((ƒ(y)-m)2*d(y))

Volumes of a Solid of Revolution: Washer Method

If a defined region, bounded by a differentiable function ƒ, on a graph is rotated about a line, the resulting solid is called a solid of revolution and the line is called the axis of revolution. The washer method is used when the defined region has space between the axis of revolution on the interval [a,b]

Revolving around the x – axis, where ƒ(x)≥g(x) (meaning the function ƒ is always above the function g on the graph) for every x on the interval [a,b]

Volume =π*(∫_a^b)(([ƒ(x)2-g(x)2])*d(x))

Revolving around the y – axis, where ƒ(y)≥g(y)(meaning the function ƒ is always to the right of the function g on the graph)

Volume =π*(∫_a^b)(([ƒ(y)]2-[g(y)]2)*d(y))

Revolving around a horizontal line y=k, where ƒ(x)≥g(x)(meaning the function ƒ is always above the function g on the graph) for every x on the interval [a,b]

Volume =π*(∫_a^b)(([ƒ(x)-k]2-[g(x)-k]2)*d(x))

Revolving around a vertical line x=m, where ƒ(y)≥g(y)(meaning the function ƒ is always to the right of the function g on the graph)

Volume =π*(∫_a^b)(([ƒ(y)-m]2-[g(y)-m]2)*d(y))

Volumes of Known Cross Sections

If a defined region, bounded by a differentiable function ƒ, is used at the base of a solid, then the volume of the solid can be found by integrated using known area formulas.

For the cross sections perpendicular to the x – axis and a region bounded by a function ƒ , on the interval [a,b], and the axis.

Cross sections are squares:

Volume =(∫_a^b)([ƒ(x)]2*d(x))

Cross sections are equilateral triangles:

Volume =√(,3)/4*(∫_a^b)([ƒ(x)]2*d(x))

Cross sections are isosceles right triangles with a leg in the base:

Volume =1/2*(∫_a^b)([ƒ(x)2]*d(x))

Cross sections are isosceles right triangles with the hypotenuse in the base:

Volume =1/4*(∫_a^b)([ƒ(x)]2*d(x))

Cross sections are semicircles (with diameter in base)

Volume =π/8*(∫_a^b)([ƒ(x)]2*d(x))

Cross sections are semicircles (with radius in base)

Volume =π/2*(∫_a^b)([ƒ(x)]2*d(x))

Differential Equations

A differential equation is an equation involving an unknown function and one or more of its derivatives

d(y)/d(x)=ƒ(x,y) → Usually expressed as a derivative equal to an expression in terms of x and or y

To solve differential equations, use the technique of separation of variables.

BC Only: Euler's Method for Approximating the Solution of a Differential Equation

Euler’s method uses a linear approximation with increments (steps), h, for approximating the solution to a given differential equation, d(y)*d(x)=F(x,y), with a given initial value.

Process: Initial Value ((x_0),(y_0))

(x_1)=(x_0)+h, (y_1)=(y_0)+h⋅F((x_0),(y_0))

(x_2)=(x_1)+h, (y_2)=(y_1)+h⋅F((x_1),(y_1))

(x_3)=(x_2)+h, (y_3)=(y_2)+h⋅F((x_2),(y_2))

This process repeats until the desired y – value is given.

Slope Field

The derivative of a function gives the value of the slope of the function at each point (x,y). A slope field is a graphical representation of all of the possible solutions to a given differential equation. The slope field is generated by plugging in the coordinates of every point (x,y) into the differential equation and drawing a small segment of the tangent line at each point.