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.
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.