Definite Integral - Area Between Two Different Curves Example

Defining the Definite Integral

The definite integral is a fundamental concept in calculus.
It represents the signed area under the curve within a specified interval.
It measures the accumulation of quantities over an interval.

Notation of Definite Integral

The definite integral of a function f(x) over the interval [a, b] is denoted as ∫[a, b] f(x) dx.
Here, f(x) is the integrand, and dx represents the differential of x.
The limits of integration, a and b, define the interval over which integration is performed.

Properties of the Definite Integral

Linearity: ∫[a, b] (cf(x) + dg(x)) dx = c∫[a, b] f(x) dx + d∫[a, b] g(x) dx
Additive Property: ∫[a, b] f(x) dx + ∫[b, c] f(x) dx = ∫[a, c] f(x) dx
Symmetry: ∫[a, b] f(x) dx = -∫[b, a] f(x) dx

Evaluation Techniques for Definite Integral

Riemann Sums: Approximating the definite integral by dividing the interval into small subintervals and summing the areas of rectangles.
Antiderivatives: If the integrand has an antiderivative, the definite integral can be evaluated using the Fundamental Theorem of Calculus.
Substitution: Changing the variable of integration to simplify the integral.

Example: Riemann Sums

Consider the function f(x) = x^2 on the interval [0, 1].
Divide the interval into 4 subintervals of equal width: [0, 1/4, 1/2, 3/4, 1].
Choose sample points within each subinterval: [1/8, 3/8, 5/8, 7/8].
Calculate the sum of the areas of the rectangles: (1/8)^2 + (3/8)^2 + (5/8)^2 + (7/8)^2.

Example: Antiderivative

Evaluate the definite integral ∫[0, 2] 2x dx.
Find the antiderivative of 2x: F(x) = x^2.
Substitute the limits of integration: F(2) - F(0).
Evaluate the definite integral: (2^2) - (0^2) = 4.

Example: Substitution

Evaluate the definite integral ∫[-1, 1] x^3 sqrt(1 - x^2) dx.
Make the substitution u = 1 - x^2.
Calculate du/dx and dx/du.
Substitute x and dx in terms of u.
Rewrite the integral with the new variable: ∫[u(-1), u(1)] -sqrt(u) du.
Evaluate the definite integral using antiderivatives.

Applications of the Definite Integral

Area under a curve: Calculating the area enclosed by a curve and the x-axis within a specified interval.
Volume of revolution: Finding the volume of a solid obtained by rotating a curve about a line.
Average value: Determining the average value of a function over an interval.

Example: Area under a Curve

Find the area enclosed by the curve y = x^2, the x-axis, and the lines x = 0 and x = 1.
Evaluate the definite integral ∫[0, 1] x^2 dx.
Use the antiderivative to find the area: (1^3)/3 - (0^3)/3 = 1/3.

Summary

The definite integral represents the signed area under the curve within a specified interval.
It is denoted as ∫[a, b] f(x) dx, where f(x) is the integrand and [a, b] are the limits of integration.
Various techniques can be used to evaluate definite integrals: Riemann sums, antiderivatives, and substitution.
The definite integral has applications in calculating areas, volumes, and average values.

Definite Integral - Area Between Two Different Curves Example

Consider the functions f(x) = x^2 and g(x) = x.
Find the area between the curves f(x) and g(x) from x = 0 to x = 1.
Set up the integral ∫[0, 1] (f(x) - g(x)) dx.
Simplify the equation: x^2 - x.
Evaluate the definite integral: [(1^3)/3 - (0^3)/3] - [(1^2)/2 - (0^2)/2].

Definite Integral - Volume of Revolution Example

Find the volume of the solid obtained by rotating the curve y = x^2 about the x-axis from x = 0 to x = 1.
Use the formula V = ∫[0, 1] πf(x)^2 dx.
Substitute the function f(x) = x^2 into the formula.
Evaluate the definite integral: π[(1^5)/5 - (0^5)/5].

Definite Integral - Average Value Example

Find the average of the function f(x) = x^2 over the interval [0, 2].
Use the formula Avg = [∫[0, 2] f(x) dx] / (2 - 0).
Substitute f(x) = x^2 into the formula.
Evaluate the definite integral: [(2^3)/3 - (0^3)/3] / 2.

Definite Integral - Integration by Parts

Integration by parts is a technique used to evaluate certain types of integrals.
The formula for integration by parts is ∫ u dv = uv - ∫ v du.
Choose u and dv from the integrand.
Calculate du and v by differentiating and integrating.
Substitute these values into the formula and evaluate the integral.

Definite Integral - Trigonometric Substitution

Trigonometric substitution is used to evaluate integrals involving radical expressions and trigonometric identities.
Substitute x = asin(t), x = acos(t), or x = a*tan(t) to simplify the integral.
Use trigonometric identities to rewrite the integral in terms of trigonometric functions.
Differentiate the substitution equation to find dx.
Substitute these values into the integral and evaluate.

Definite Integral - Partial Fractions

Partial fractions is a method used to decompose a rational function into simpler fractions.
Factorize the denominator of the rational function.
Set up the partial fractions as A/(factor 1) + B/(factor 2) + … + N/(factor n).
Multiply both sides by the common denominator.
Equate the coefficients of the corresponding powers of x and solve for the unknowns.

Definite Integral - Improper Integrals

Improper integrals are used to evaluate integrals with infinite limits or discontinuities.
Type 1: The upper or lower limit of integration is infinite.
Type 2: The integrand is discontinuous within the interval.
Evaluate the improper integral as a limit as the limiting value approaches infinity or the point of discontinuity.

Definite Integral - Comparison Test

The comparison test is used to evaluate improper integrals by comparing them to known convergent or divergent series.
If 0 ≤ f(x) ≤ g(x) for all x in the interval, and ∫[a, b] g(x) dx is convergent, then ∫[a, b] f(x) dx is also convergent.
If 0 ≤ g(x) ≤ f(x) for all x in the interval, and ∫[a, b] g(x) dx is divergent, then ∫[a, b] f(x) dx is also divergent.

Definite Integral - Limit Comparison Test

The limit comparison test is an alternative method to evaluate improper integrals.
If lim(x->∞) (f(x) / g(x)) = L, where L is a finite positive number, then both integrals either converge or diverge together.
Choose a function g(x) with convergent or divergent properties.
Divide f(x) by g(x) and take the limit as x approaches the specified value.
Determine the convergence or divergence based on the result.

Definite Integral - Summary

Various techniques can be used to evaluate definite integrals, including integration by parts, trigonometric substitution, and partial fractions.
Improper integrals are used when the limits of integration are infinite or the integrand has discontinuities.
The comparison test and limit comparison test are methods to determine the convergence or divergence of improper integrals.
Understanding these techniques and concepts will enable you to solve a wide range of integration problems.

Definite Integral - Integration by Substitution

Integration by substitution is a technique used to simplify integrals by changing the variable of integration.
Choose a substitution that will simplify the integral.
Rewrite the integral in terms of the new variable.
Calculate the differential of the new variable.
Substitute the new variable and its differential into the integral.
Evaluate the integral using antiderivative techniques.

Definite Integral - L’Hopital’s Rule

L’Hopital’s Rule is used to evaluate limits involving indeterminate forms.
The indeterminate forms are 0/0 and ∞/∞.
If lim(x->a) f(x) / g(x) is in an indeterminate form, where a is a real number, then apply L’Hopital’s Rule.
Take the derivative of the numerator and denominator separately.
Evaluate the limit of the derivatives.
If the limit still remains in an indeterminate form, repeat the process until a determinate form is obtained.

Definite Integral - Hyperbolic Trigonometric Substitution

Hyperbolic trigonometric substitution is a technique used to simplify integrals involving rational functions and square roots.
Use the substitutions x = sinh(t), x = cosh(t), or x = tanh(t) to transform the integral.
Rewrite the integral using hyperbolic trigonometric identities.
Calculate dx in terms of dt.
Substitute the new variables and differentials into the integral.
Evaluate the integral using antiderivative techniques.

Definite Integral - Series Expansion

Series expansion is used to approximate integrals when finding an exact solution is difficult.
Express the function or integrand as a power series expansion.
Choose the appropriate number of terms in the series to achieve the desired level of accuracy.
Integrate each term of the series expansion.
Sum the integrals of each term to obtain the approximate value of the integral.

Definite Integral - Double Integral

Double integrals are used to calculate the volume, area, and mass of two-dimensional regions in space.
The general form of a double integral is ∬R f(x, y) dA, where R is a region in the xy-plane.
Evaluate the inner integral first and treat all other variables as constants.
Evaluate the outer integral by using the limits of integration determined by the region R.

Definite Integral - Triple Integral

Triple integrals extend the concept of the definite integral to three-dimensional space.
The general form of a triple integral is ∭D f(x, y, z) dV, where D is a region in three-dimensional space.
Evaluate the innermost integral first and treat all other variables as constants.
Evaluate the intermediate integrals by using the limits of integration determined by the region D.
Evaluate the outer integral to obtain the final result.

Definite Integral - Green’s Theorem

Green’s Theorem relates a line integral around a simple closed curve C to a double integral over the plane region R bounded by C.
The general form of Green’s Theorem is ∮C F • dr = ∬R (curl F) • dA.
F is a vector field, dr is an infinitesimal displacement vector, and dA is an infinitesimal area element.
Evaluate the line integral by parameterizing the curve C and calculating the dot product between F and dr.
Evaluate the double integral by determining the curl of F and integrating over the region R.

Definite Integral - Stokes’ Theorem

Stokes’ Theorem relates a surface integral of a vector function to a line integral around the boundary of the surface.
The general form of Stokes’ Theorem is ∬S (curl F) • dS = ∮C F • dr.
F is a vector field, dS is an infinitesimal surface element, and dr is an infinitesimal displacement vector along C.
Evaluate the surface integral by parameterizing the surface S and calculating the dot product between the curl of F and the surface normal vector.
Evaluate the line integral by parameterizing the boundary curve C and calculating the dot product between F and dr.

Definite Integral - Divergence Theorem

The Divergence Theorem relates a volume integral of a vector function to a surface integral over the boundary of the volume.
The general form of the Divergence Theorem is ∭V (div F) dV = ∬S F • dS.
F is a vector field, dV is an infinitesimal volume element, and dS is an infinitesimal surface element.
Evaluate the volume integral by parameterizing the volume V and calculating the divergence of F.
Evaluate the surface integral by parameterizing the surface S and calculating the dot product between F and dS.

Definite Integral - Summary

Integration by substitution, L’Hopital’s Rule, hyperbolic trigonometric substitution, series expansion, and other techniques are used to evaluate integrals.
Double and triple integrals are used to calculate volumes, areas, and masses in multiple dimensions.
Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem provide relationships between line integrals, surface integrals, and volume integrals.
Understanding these advanced integration techniques and concepts will allow you to solve more complex mathematical problems.